European Journal of Clinical Nutrition (2015), 1–7 © 2015 Macmillan Publishers Limited All rights reserved 0954-3007/15 www.nature.com/ejcn
ORIGINAL ARTICLE
Diet models with linear goal programming: impact of achievement functions JC Gerdessen1 and JHM de Vries2 BACKGROUND/OBJECTIVES: Diet models based on goal programming (GP) are valuable tools in designing diets that comply with nutritional, palatability and cost constraints. Results derived from GP models are usually very sensitive to the type of achievement function that is chosen. This paper aims to provide a methodological insight into several achievement functions. It describes the extended GP (EGP) achievement function, which enables the decision maker to use either a MinSum achievement function (which minimizes the sum of the unwanted deviations) or a MinMax achievement function (which minimizes the largest unwanted deviation), or a compromise between both. An additional advantage of EGP models is that from one set of data and weights multiple solutions can be obtained. SUBJECTS/METHODS: We use small numerical examples to illustrate the ‘mechanics’ of achievement functions. Then, the EGP achievement function is demonstrated on a diet problem with 144 foods, 19 nutrients and several types of palatability constraints, in which the nutritional constraints are modeled with fuzzy sets. RESULTS: Choice of achievement function affects the results of diet models. CONCLUSIONS: MinSum achievement functions can give rise to solutions that are sensitive to weight changes, and that pile all unwanted deviations on a limited number of nutritional constraints. MinMax achievement functions spread the unwanted deviations as evenly as possible, but may create many (small) deviations. EGP comprises both types of achievement functions, as well as compromises between them. It can thus, from one data set, find a range of solutions with various properties. European Journal of Clinical Nutrition advance online publication, 22 April 2015; doi:10.1038/ejcn.2015.56 INTRODUCTION Mathematical modeling of diets can be defined as the use of mathematical techniques to formulate and optimize diets.1 We refer to Buttriss et al.1 for a description of relevance, history and applications of diet (planning) models based on linear programming (LP). LP-based diet models contain decision variables, an objective function and a set of constraints. Commonly, the decision variables are defined as: Xi = (proposed) daily intake of food i (i = 1...I), Xi ⩾ 0. The objective function minimizes (or maximizes) a linear function of the decision variables, for example total cost or total energy content of the diet. The (linear) constraints ensure that the proposed diet meets requirements on, for example, nutrient content and palatability. If the model contains integer variables, it is called a mixed integer LP model. Classification of diet models We distinguish between two classes: 1. Single-objective problems: nP minimize o (or maximize) one linear I function of Xi: Minimize i¼1 ci X i . If ci represents the cost of food i, then the objective function minimizes total diet cost and so on.2–4 Commonly, single-objective problems are formulated as straightforward (MI)LP models. 2. Multi-objective problems: two types are observed in literature: 2a. With the set of available foods no diet can be planned that complies with all constraints.5–7 A common approach is to search for a ‘best possible’ diet, which violates the
constraints as little as possible. This is a problem with multiple objectives: Minimize{Violation of constraint 1}, Minimize{Violation of constraint 2}, …, Minimize{Violation of the last constraint}. 2b. The decision maker aims to plan a diet that complies with all constraints, and that differs as little as possible from the actual diet (which does not comply with all constraints).4,8–11 In other words, (s)he wants to minimize the differences between the proposed (optimized) diet XP and the actual diet XA: Minimize X A1 - X P1 ; Minimize X A2 - X P2 ; :::; Minimize X AI - X PI :
This paper focuses on multi-objective diet models in Class 2a. For completeness, Class 2b is discussed in Supplementary Appendix A.
Multi-objective problems—searching Pareto-optimal solutions The objectives of a multi-objective problem are usually conflicting; commonly no solution exists that optimizes all objectives at the same time. For instance, the diet with the lowest violation of a constraint on iron intake might have a considerable violation of a constraint on intake of saturated fat, and vice versa. Methods for generating solutions for multi-objective problems therefore commonly focus on finding so-called Pareto-optimal solutions (also denoted as efficient solutions). Solution X is called Paretooptimal if no other solution X# exists that performs at least as good as X with respect to all objectives, and better with respect to at least one objective.12 In other words, the achieved value for one
1 Group Operations Research and Logistics, Wageningen University, Wageningen, The Netherlands and 2Division of Human Nutrition, Wageningen University, Wageningen, The Netherlands. Correspondence: JC Gerdessen, Operations Research and Logistics, Wageningen University, Hollandseweg 1, Wageningen 6706 KN, The Netherlands. E-mail: [email protected] Received 2 November 2014; revised 25 February 2015; accepted 26 February 2015
Achievement functions in diet models JC Gerdessen and JHM de Vries
2 objective cannot be improved without worsening the level of another objective.13 The concept of Pareto-optimality is illustrated in Table 1, which shows four fictitious diets, and their violations vFe and vSF of intake constraints on iron and saturated fat, respectively. No diet exists that has both lowest vFe and lowest vSF. We therefore aim to identify the Pareto-optimal diets. All references mentioned in Class 2 use linear goal programming (GP). GP uses the following steps12 to find ‘best possible’ (that is, Pareto-optimal) diets for problems in Class 2a: ● ●
(GP1): quantify the extent to which a diet violates the constraints. (GP2): minimize a function of these violations in order to obtain a diet that violates the constraints as little as possible. This function is called the achievement function.
Multi-criteria decision-making literature, in which GP is positioned, recognizes achievement function selection, weight selection and weight space analysis as topics of major importance for the quality of decision making,13–16 because any choice made in formulating the achievement function uses judgment of the modeler and implies assumptions on the preference structure of the decision maker. If the election of the achievement function is wrong, then the decision maker will probably not accept the solution.15 Results derived from GP models usually are very sensitive to the type of achievement function that is chosen.15 All references mentioned in Class 2 use a weighted additive achievement function. GP literature offers other achievement functions as well.15 Typically, with a weighted additive achievement function, each set of data and weights results in a single solution. It would be useful to offer the decision maker more than just this single Table 1.
A B C D
MATERIALS AND METHODS Goal programming Numerical example. In order to reveal the ‘mechanics’ of achievement functions, we use a simplified diet model with two foods: (1) bread and (2) meat, with associated decision variables X1 (X2) = amount of bread (meat) in the diet. An upper bound on salt intake restricts bread consumption to three units or less (nutritional constraint, Eq. (1)). An upper bound on saturated fat intake restricts meat consumption to two units or less (nutritional constraint, Eq. (2)). For sufficient iron intake the diet should contain at least six units of bread and/or meat (nutritional constraint, Eq. (3)): X1 X2 X1 þ X2 X1; X2 ⩾ 0
ð1Þ ð2Þ ð3Þ
Iron: vFe
Saturated fat: vSF
0 4 6 3
11 3 4 8
Pareto-optimal or not?
Yes Yes No: B has lower vFe and vSF Yes
3 ð1Þ 2 ð2Þ 6 ð3Þ
⩽ 3 ⩽ 2 ⩾ 6
X1 X2 þ X2
X1
- X1 - X1
þd1þd2þd3-
X2 X2
þ
- dþ 1 - dþ 2 - dþ 3
¼ 3 ¼ 2 ¼ 6
For instance, (X1,X2) = (1,1) implies d1- ¼ 2; d 2- ¼ 1; d 3- ¼ 4 and (X1,X2) = (6,1) þ implies d þ 1 ¼ 3; d 2 ¼ 1; d 3 ¼ 1. (Note that in any optimal solution either one of the deviational variables (d–j or d+j ) equals zero, or both.12 ) As initial constraint (3) provides a lower bound for X1+X2, it is allowed to have a positive deviation d þ 3 . Any shortage d 3 , however, violates nutritional constraint (3) and is therefore considered an unwanted deviation. Likewise, þ dþ 1 and d 2 are unwanted deviations, see Figure 1b. Next, step (GP2) searches the ‘best possible’ diet by formulating and minimizing the achievement function.
b 4
X2 →
X2 →
a
⩽ ⩽ ⩾
Figure 1a shows that constraints (1)–(3), are conflicting: no diet (X1, X2) exists that complies with all constraints. The resulting model belongs to Class 2a. Step (GP1) quantifies the violation of the nutritional constraints by adding deviational variables dj- ; dþ j ⩾0, which represent the negative and positive deviation from the right-hand side value of nutritional constraint j:
Example of Pareto-optimality Violation of intake constraint on
Diet Diet Diet Diet
solution and present a range of Pareto-optimal solutions. Offering multiple solutions allows choice of a solution which is most suitable for a specific decision problem, and that best meets nonquantifiable goals and preferences.17–19 This paper aims to provide methodological insight into several GP achievement functions: MinSum, MinMax and extended GP (EGP). It shows that the EGP achievement function is able to generate a solution that minimizes the sum of all violations, as well as a solution that minimizes the largest violation, and compromises between them. The EGP achievement function thus provides a way to obtain a range of solutions from one set of data and weights.
4
3
3
d2+ 2
2
––
6 X1 →
–
5
d3
4
6
3
+
+=
X1+ d1––d1+= 3 0
d3
d3
2
d1+
d3
1
+
d1–
X1≤ 3 0 0
(6,1)
2
(1,1) 1
6
1
d2–
+X X1
≥ 2 +X X1
X2≤ 2
X2+ d2––d2+= 2
0
1
2
3
4
5
6 X1 →
Figure 1. (a) A diet model in Class 2a. No diet exists that complies with all constraints. (b) A GP model uses deviational variables to quantify þ the deviations from the targets. Deviations d 1- ; d 2- ; d þ ) are allowed by the original constraints. The dþ 3 ( 1 ; d 2 ; d 3 ( → ) are unwanted deviations. A MinSum GP model will propose a corner point (•) of the shaded area, no matter which set of weights is used. A MinMax GP model is able to propose solutions inside the shaded area, e.g., if all weights are equal then (X1,X2) = (31/3, 21/3) (m) is proposed (constraints, Eqs. (14)–(16), cannot be visualized in two dimensions. Therefore it cannot visualized in b that point (X1, X2) = (31/3, 21/3) is a corner point in the nine-dimensional space (X1, X2, Dsum, d–1, d+1 , d–2, d+2 , d–3, d+3 ). European Journal of Clinical Nutrition (2015) 1 – 7
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Achievement functions in diet models JC Gerdessen and JHM de Vries
3 MinSum achievement function. A weighted additive achievement function —which we refer to as MinSum achievement function—minimizes a weighted sum of the unwanted deviations. For model (1)–(3), the þ unwanted deviations are d þ 1 ; d 2 ; d 3 , which results in the following MinSum GP model:
●
●
þ þ þ - Minimize Dsum ¼ w þ 1 d 1 þ w 2 d 2 þ w 3 d 3 ð4Þ þ þ d1 - d1 ¼ 3 ð5Þ X1 X 2 þ d2- - dþ ¼ 2 ð6Þ 2 X 1 þ X 2 þ d3- - dþ ¼ 6 ð7Þ 3 þ þ ð8Þ X 1 ; X 2 ; Dsum ; d 1- ; dþ 1 ; d2 ; d2 ; d3 ; d3 ⩾ 0 þ in which w þ 1 ; w 2 ; w 3 are user-defined, non-negative weights. The optimal solution of problem, Eqs. (4–8), depends on the weights. Independent of the weights, it is always located in the shaded area of Figure 1b. For þ wþ 1 ¼ w 2 ¼ w 3 all solutions in the shaded area have the same value of Dsum. As in LP-problems the optimal solution is always a corner point,12 an LP-solver will generate a corner point of the shaded area: (3,2) or (4,2) þ or (3,3). However, if the user specifies w þ 1 ¼ w 2 ¼ w 3 (s)he expresses that (s)he does not want to prioritize one nutrient, that is, (s)he cannot justify assigning one deviation more importance than another. Hence, it would be natural to obtain a balanced solution that spreads theþ total unwanted deviation over all three deviational variables dþ 1 ; d2 ; d3 instead of an unbalanced solution that piles the unwanted deviation on only one of them. Table 2 shows that the MinSum model is sensitive to weight changes: slight weight changes make the optimal diet ‘jump’ from one corner point to another. However, if the weights expressed by the user are similar, one would expect similar solutions. These imbalance and sensitivity are typical for an additive achievement function.12,14 Using an additive achievement function implies the assumption that all weighted unwanted deviations are additive and that nutritional adequacy of a diet is determined by the sum of its weighted unwanted deviations. This presupposes—implicitly—that
Table 2.
●
deviations can compensate each other, for example, an increase in the deviation from a vitamin C target can be compensated by a decrease in the deviation from a calcium target; tradeoffs between the deviations are precisely known, for example, 10% deviation from a vitamin C target is considered equally ‘grave’ as 5% deviation from a calcium target; and tradeoffs are constant and do not depend on intake level, for example, 10% extra deviation from a vitamin C target is considered equally ‘grave’ as 5% extra deviation from a calcium target, no matter whether the vitamin C intake is almost adequate or dangerously low.
MinMax achievement function. A MinMax achievement function (also called Chebyshev achievement function) aims to minimize the largest among the weighted unwanted deviations:14,15,20 þ þ þ - Minimize Dmax ¼ max w þ ð9Þ 1 d1 ; w 2 d2 ; w 3 d3 In order to obtain a linear model that minimizes this non-linear þ achievement function a constraint Dmax ⩾ w j- dj- ðDmax ⩾ w þ j d j Þ is added 21 for every unwanted deviation dj- ðdþ j Þ Minimize fDmax g þ d1- - dþ ¼ 3 X1 1 X 2 þ d2- - dþ ¼ 2 2 X 1 þ X 2 þ d3- - dþ ¼ 6 3 þ Dmax ⩾ w þ 1 d1 þ Dmax ⩾ w þ 2 d2 Dmax ⩾ w 3- d 3þ þ X 1 ; X 2 ; Dmax ; d 1- ; dþ 1 ; d2 ; d2 ; d3 ; d3 ⩾ 0
ð10Þ ð11Þ ð12Þ ð13Þ ð14Þ ð15Þ ð16Þ ð17Þ
Achievement function (10) and constraints together ensure that (14–16) þ þ þ - Dmax will take the value Dmax ¼ max w þ 1 d 1 ; w 2 d 2 ; w 3 d 3 , and that Dmax is as low as possible. Table 2 shows that using the MinMax achievement function for þ wþ 1 ¼ w 2 ¼ w 3 ¼ 1 results in a balanced solution: all unwanted
Solutions of the MinSum, MinMax and extended GP model Optimal solution of MinSum GP model (4)–(8) for various choices of the weights
Weights þ wþ 1 ; w2 ; w3 (1, 1, 1) (0.9, 1, 1) (1, 0.9, 1) (1, 1, 0.9)
Diet (X1,X2) (4,2) or (3,3) or (3,2) (4, 2) (3, 3) (3, 2)
þ Deviations dþ 1 ; d2 ; d3
deviations Weighted þ þ þ - wþ 1 d1 ; w2 d2 ; w 3 d3
Achievement function þ þ þ - Dsum ¼ w þ 1 d1 þ w2 d2 þ w 3 d3
(1,0,0) or (0,1,0) or (0,0,1)
(1,0,0) or (0,1,0) or (0,0,1)
1
(1, 0, 0) (0, 1, 0) (0, 0, 1)
(0.9, 0, 0) (0, 0.9, 0) (0, 0, 0.9)
0.9 0.9 0.9
Optimal solution of MinMax GP model (10)–(17) for various choices of the weights Weights þ wþ 1 ; w2 ; w3 (1, 1, 1) (0.9, 1, 1) (1, 0.9, 1) (1, 1, 0.9)
Diet (X1,X2) (3.333, (3.357, (3.321, (3.321,
2.333) 2.321) 2.357) 2.321)
þ Deviations dþ 1 ; d2 ; d3 (0.333, (0.357, (0.321, (0.321,
0.333, 0.321, 0.357, 0.321,
0.333) 0.321) 0.321) 0.357)
Achievement function þ þ þ - Dmax ¼ max w þ 1 d1 ; w 2 d2 ; w3 d3
deviations Weighted þ þ þ - wþ 1 d1 ; w 2 d2 ; w 3 d3 (0.333, (0.321, (0.321, (0.321,
0.333, 0.321, 0.321, 0.321,
0.333) 0.321) 0.321) 0.321)
0.333 0.321 0.321 0.321
Optimal solution of extended GP þmodel (11)–(18) for various choices of parameter λ. Weights w þ 1 ; w 2 ; w 3 ¼ ð0:5; 0:75; 1Þ were used. Parameter λ
Diet (X1,X2)
0.00 0.25 0.50 0.75 1.00
(4, (4, (3.60, (3.46, (3.46,
2) 2) 2.40) 2.31) 2.31)
þ Deviations d þ 1 ; d2 ; d3 (1, 0, 0) (1, 0, 0) (0.60, 0.40, 0) (0.46, 0.31, 0.23) (0.46, 0.31, 0.23)
deviations Weighted þ þ þ - wþ 1 d1 ; w 2 d2 ; w 3 d3
Dsum
Dmax
Dext = (1 − λ)Dsum+λ Dmax
(0.50, 0, 0) (0.50, 0, 0) (0.30, 0.30, 0) (0.23, 0.23, 0.23) (0.23, 0.23, 0.23)
0.50 0.50 0.60 0.69 0.69
0.50 0.50 0.30 0.23 0.23
0.50 0.50 0.45 0.35 0.23
Abbreviation: GP, goal programming.
© 2015 Macmillan Publishers Limited
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Achievement functions in diet models JC Gerdessen and JHM de Vries
4 deviations are equal. Moreover, the model is less sensitive to weight changes: slight weight changes cause minor shifts in (X1,X2). The MinMax achievement function attempts to spread the unwanted deviations as evenly as possible and it enables to identify non-corner point solutions in the shaded area. However, possibly many unwanted deviations are found.14 With a MinMax achievement function the nutritional adequacy of a diet is mainly determined by its poorest nutrient. Extended GP achievement function. The MinSum and MinMax achievement functions can be combined into the so-called EGP achievement function Dext:14,15 Minimize fDext ¼ ð1 - λÞUDsum þ λUDmax g
ð18Þ
where parameter λ ∈ (0;1) weighs the importance attached to minimization of Dsum and Dmax. Dext comprises both Dsum and Dmax: using λ = 0 implies Dext = Dsum, so the model minimizes the weighted sum of the unwanted deviations. Using λ = 1 implies Dext = Dmax, so the model spreads the deviations and keeps the largest unwanted deviation as low as possible. For intermediate values of λ solutions are found that are compromises between the MinSum and MinMax solution. Thus, by varying λ the decision maker can obtain a set of dietary suggestions ‘between’ the MinSum and MinMax diets, see also Table 2. Normalization to (0;1)-interval; interpretation as fuzzy sets. In order to deal with issues of scaling and incommensurability it is useful to formulate a GP model in such a way that unwanted deviational variables are automatically normalized to a (0;1) interval.13 For instance, consider a diet model with two foods and two nutrients, see Table 3.
Definition dl j- ; dl þ j deviations from the left bound of target intake for nutrient j: dr j- ; dr þ j deviations from the right bound of target intake for nutrient j:
The dl j- and dr þ j are the unwanted deviations. The nutritional constraints can be formulated as 500X 1
þ
800X 2
þ
500X 1
þ
800X 2
þ
0:3X 1
þ
0:1X 2
þ
0:3X 1
þ
0:1X 2
þ
500dl 1dl 14000dr 1-
-
500dl þ 1
-
0:3dl 2dl 22:6dr 2-
-
4000dr þ 1 dr þ 1 0:3dl þ 2
-
2:6dr þ 2 dr þ 2
¼ 1500 ð19Þ ⩽ 1 ð20Þ ¼ 3000 ð21Þ ⩽ 1 ð22Þ ¼ 0:9 ð23Þ ⩽ 1 ð24Þ ¼ 2:4 ð25Þ ⩽ 1 ð26Þ
Thus, all unwanted deviational variables are normalized to [0;1], which facilitates judgment of tradeoffs. Moreover, the normalized deviational variables can be used to incorporate fuzzy sets for nutrient intake. Figure 2 shows a graph of intake I1 of nutrient 1 versus – + μ1 ¼ 1 - dl 1- - dr þ 1 . (In an optimal solution dlj and drj can never be both non-zero at the same time, so μ1 ðI1 Þ ¼ min 1 - dl 1- ; 1 - dr þ 1 ¼ 1 - dl 1- - dr þ 1 .) For 1500 ⩽ I1 ⩽ 3000 the intake of nutrient 1 is considered þ fully adequate: both unwanted deviational variables dl 1 and dr 1 are zero and μ1 ¼ 1: Intake I1 = 1000 is considered fully inadequate: dl 1- ¼ 1 and μ1 = 0. Likewise, I1 = 7000 has dr þ 1 ¼ 1 and μ1 = 0. So an adequate intake has μ1 = 1 and an inadequate intake has μ1 = 0. If we assume the adequacy of intake I1 increases linearly from 0 to 1 in interval (1000;1500), we can use μ1 as proxy for the adequacy of the diet with respect to nutrient 1. In case overall nutritional quality Μ of the diet is determined by the adequacy of its poorest nutrient we can calculate it as
Table 3.
þ M ¼ Minimumfμ1 ; μ2 g ¼ 1 - Maximum dl 1- ; dr þ ¼ 1 - Dmax , 1 ; dl 2 ; dr 2 which implies the MinMax achievement function yields diets with maximal M. The adequacy curve in Figure 2 can be interpreted as fuzzy set for the adequacy of intake I1 with membership function μ1 ðI1 Þ ¼ 1 - dl 1- - dr þ 1 . For more information on use of fuzzy sets for modeling intake adequacy we refer to Gedrich et al.,22 Wirsam et al.,23 Wirsam and Uthus.24 More information on LP-formulations for fuzzy sets is found in Yaghoobi and Tamiz.25 Mixed Integer LP-formulations of Dantzig26 can be used to construct curves with more than three intervals.
RESULTS The EGP achievement function is used to obtain ‘best possible’ solutions for a diet model for planning of diets for dietary controlled trials for men aged 19–30y. Palatability constraints define lower and upper bounds on intake of various foods (‘at most 245 g of bread’), and they link intakes of foods (‘3–7 g spread per slice’). Nutritional constraints are formulated via adequacy curves for 19 nutrients and for vegetables and fruits. The four characteristic points of each adequacy curve are defined in the following way: a is the Lower Intake Level below which an intake could lead to risk in most individuals; b is the average requirement, sufficient for virtually 50% of healthy people in a group, c is the recommended daily intake which is sufficient for nearly all people; d is the upper intake level that is unlikely to pose a risk of adverse health effects.27 If no information was available a nutrition expert (JdV) made an estimate.27,28 Energy intake was fixed to 100% of the estimated average requirements. Supplementary Appendix B provides the full model. The model was programmed in Fico Xpress 7.0.1 (Fair Isaac Corporation, London, UK) and calculations were done on an HP desktop (Hewlett-Packard Company, Palo Alto, CA, USA) with Intel I7 processor. The model generates the 11 diets in one second. Table 4 provides reference values for the adequacy curves and summarizes results for λ ∈ {0;0.1;…;1}. Using λ = 0 implies Dext = Dsum, so the model finds the diet with minimal sum of unwanted deviations. In this MinSum diet all nutrient intakes except mono- and disaccharides (MDSacch) are in their optimal range (at plateau [b,c] of the adequacy curve). However, this diet has dr þ MDSacch ¼ ð7:30 - 5Þ=ð10 - 5Þ ¼ 0:460, which means that MDSacch intake is far too high and has a suboptimal adequacy μMDSacch = 1 − 0.460 = 0.540. As MDSacch is the only nutrient with
Figure 2.
Adequacy curve for nutrient 1.
Data for diet model Nutrient content
Nutrient 1 Nutrient 2
Nutritional constraints
(mg in 100 g of food 1)
(mg in 100 g of food 2)
Lower bound on intake (mg)
Target intake (mg)
Upper bound on intake (mg)
500 0.3
800 0.1
1000 0.6
(1500; 3000) (0.9; 2.4)
7000 5
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© 2015 Macmillan Publishers Limited
en% en% en% en% en% en% mg mg en% gr/MJ mg mg mg mg mg mg μg mg μg g g
Protein Total fat SFA MUFA PUFA Linol EPA+DHA Cholesterol MDSacch Fiber Ca Fe K Vit B1 Vit B2 Vit B6 Vit B12 Vit C Folate Vegetables Fruits
8% 2% 0% 8% 0% 0% 350 0 0% 0 400 7 1600 0.5 0.8 0.8 1 10 100 150 100
a
10% 25% 0% 10% 3% 2% 450 0 0% 3 500 9 3100 0.9 1.1 1 1.4 50 200 200 200
b 20% 40% 10% 20% 10% 9% 3000 200 5% —b 800 23 3500 1.4 1.7 1.5 2 75 400 400c 300c
c 25% 45% 15% 28% 12% 10% 4000 300 10% —b 2500 60 10 000 7 8.5 25 10 375 2000 400c 300c
d 14.0% 40.0% 10.0% 17.6% 10.0% 8.82% 450 74.6 7.30% 3.00 720 12.1 3227 1.35 1.1 1.5 2.0 75 292 200 200 Dsum Dmax Dext
I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
dl − 0 0 0 0 0 0 0 0 0.460 0 0 0 0 0 0 0 0 0 0 0 0
dr+ 1 1 1 1 1 1 1 1 0.540 1 1 1 1 1 1 1 1 1 1 1 1 0.460 0.460 0.460
μ 1 1 1 1 1 1 1 1 0.691 1 1 1 1 1 1 1 1 0.846 1 1 1 0.463 0.309 0.448
0.1
1 1 1 1 1 1 1 1 0.692 1 1 1 1 1 1 1 0.990 0.854 1 1 1 0.463 0.308 0.432
0.2
1 1 1 1 1 1 1 1 0.700 1 1 1 1 1 1 1 0.981 0.852 1 1 1 0.466 0.300 0.416
0.3
1 1 1 1 1 1 1 1 0.702 1 1 1 1 1 1 1 0.981 0.850 1 1 1 0.468 0.298 0.400
0.4
1 1 1 1 1 1 1 1 0.703 1 1 1 1 1 1 1 0.980 0.848 1 1 1 0.469 0.297 0.383
μ
0.5
1 1 1 1 1 1 1 1 0.704 1 1 1 1 1 1 1 0.980 0.847 1 1 1 0.469 0.296 0.365
0.6
Results for λ = 0, 0.1, …, 1
1 1 1 1 1 1 1 1 0.729 1 1 1 1 1 1 0.993 0.926 0.835 1 1 1 0.518 0.271 0.345
0.7
1 1 1 1 1 1 1 1 0.782 1 1 1 1 1 1 0.993 0.911 0.866 1 1 0.782 0.666 0.218 0.307
0.8
1 1 1 1 1 1 1 1 0.820 0.820 1 1 1 0.856 1 0.979 0.869 0.867 1 1 0.820 0.969 0.180 0.259
0.9
19.2% 40.8% 10.8% 16.9% 10.3% 8.42% 434 180 5.82% 2.51 949 12.4 3745 2.32 1.26 2.15 3.31 124 309 192 184
Intake
0 0 0 0 0 0 0.163 0 0 0.163 0 0 0 0 0 0 0 0 0 0.163 0.163
dl −
1
0 0.163 0.163 0 0.163 0 0 0 0.163 0 0.088 0 0.038 0.163 0 0.028 0.163 0.163 0 0 0
dr+
1 0.837 0.837 1 0.837 1 0.837 1 0.837 0.837 0.912 1 0.962 0.837 1 0.972 0.837 0.837 1 0.837 0.837 1.95 0.163 0.163
μ
Abbreviations: EPA, eicosapentaenoic acid; DHA, docosahexaenoic acid; MDSacch, mono- and disaccharides, MUFA, mono-unsaturated fatty acid; PUFA, poly-unsaturated fatty acid; SFA, saturated fatty acid. a These units apply to a, b, c, d, I. The dl−, dr+, μ are dimensionless. bNo c and d are used, because within the feasible diets fiber intake will never be too high. cThese are palatability constraints.
unita
Nutrient
0
Reference values for nutritional adequacy curves and summary of results for λ = 0, 0.1, …, 1
Reference values for adequacy curves
Table 4.
Achievement functions in diet models JC Gerdessen and JHM de Vries
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Achievement functions in diet models JC Gerdessen and JHM de Vries
6 suboptimal intake the MinSum diet has Dmax = 0.460. Increasing λ implies that the model lowers Dmax at the cost of increasing Dsum, thus spreading the unwanted deviations in order to find a more balanced solution: for λ = 0.1 the unwanted deviation of MDSacch decreases to dr þ MDSacch ¼ 0:309 (so μMDSacch = 1 − 0.309 = 0.691) whereas vitamin C intake becomes suboptimal (μVitC = 0.846). For λ = 0.9 the largest deviation has decreased to 0.180. However, seven intakes are then suboptimal. For λ ¼ 1 the model strictly minimizes the largest unwanted deviation, which results in the diet with lowest Dmax. This MinMax diet has 14 suboptimal intakes and the highest Dsum. It is entirely up to the nutritionist to judge whether a decrease in Dmax is worth an increase in Dsum, and/or an increase in the number of violated constraints and to express a preference for any of the generated diets, based on the specific situation on hand. Supplementary Appendix C shows which foods were chosen. DISCUSSION This paper aims to provide a methodological insight into several GP achievement functions: MinSum, MinMax and EGP. A MinSum achievement function minimizes the sum of the unwanted deviations from nutritional targets and is thus appropriate in situations where diet quality is determined by the sum of these unwanted deviations. It can, however, lead to solutions that are unbalanced and that are sensitive to changes in preferential weights. A MinMax achievement function minimizes the largest among the unwanted deviations, and is thus appropriate when diet quality is mainly determined by the nutrient with the largest unwanted deviation. MinMax GP provides solutions that are as balanced as possible with respect to the unwanted deviational variables. However, possibly many unwanted deviations occur. An EGP achievement function is a compromise between MinSum and MinMax. It can—from one set of data and weights—obtain the MinSum solution, the MinMax solution, and a set of solutions ‘between’ the MinSum and MinMax solutions. Offering multiple solutions allows choice of a solution that is most suitable for a specific decision problem, and that best meets non-quantifiable goals and preferences.17–19 Extending a MinSum GP model to a MinMax GP model takes one additional variable (Dmax) plus one constraint for every unwanted deviation. In terms of model size (see also Supplementary Appendix B) this can be considered as very small. The EGP achievement function requires one extra model parameter (λ), which weighs the importance attached to minimizing the total unwanted deviation versus the importance of minimizing the largest unwanted deviation. No general rule can be given for setting the most useful value of λ. It seems most practical to let the model run for, for example, λ ∈ {0;0.1;…;1} and then judge the resulting diets and their nutritional adequacy. If desired, the model can be rerun with smaller step-sizes for λ in relevant sub-intervals, for example, λ ∈ {0.71;0.72;…;0.89}. In this way the decision maker is supported in finding his/her own tradeoffs. GP offers several other achievement functions that are worth exploring in diet modeling context. For instance, Lexicographic GP,13,15 in which the deviational variables are assigned to a number of priority levels that are minimized sequentially. In minimization runs for lower level deviational variables the higher level deviational variables are fixed to their (previously obtained) optimal values. Lexicographic GP requires the decision maker to provide a strict hierarchy of the unwanted deviations. Also, in GP it is possible to minimize the number of unmet nutritional constraints.29 This requires introducing binary variables indicating whether nutritional constraints have been met (see Supplementary Appendix D). It is useful in situations where unmet nutritional constraints incur costs, for example, due to necessary fortification programs. European Journal of Clinical Nutrition (2015) 1 – 7
A key consideration in achievement function selection is the preference structure of the decision maker.15 If the deviational variables can be classified into strict priority classes between which no finite tradeoffs exist then Lexicographic GP should be considered. If finite tradeoffs do exist between deviational variables then the decision maker should consider EGP, which offers the opportunity either to minimize the total deviation or the largest deviation or a compromise between both. A decision maker who wants to minimize the number of unmet nutritional constraints needs to introduce binary variables. It will depend on the type of nutrition question, for example whether it is aimed at the individual or population level, what the best modeling approach is. Further research is necessary to build a comprehensive framework that helps nutritionists to select the most suitable modeling approach for a wide range of diet problems. This paper focuses on linear achievement functions. In literature also models are described with quadratic achievement functions.22,30–32 In a quadratic achievement function the (weighted) sum of squared unwanted deviations is minimized, which means that large deviations are penalized more than small deviations.32 Quadratic achievement functions can find non-corner point solutions. However, possibly local optima are generated.33 The quality of the solutions of a diet model depends on the quality and choice of data.1 For our experiment, we chose a limited number of foods and nutrients, and had to base adequacy curves partly on expert opinion, because not all needed information was available in literature. Also, we did our experiment only for a specific diet of one energy level. Other foods could have been used, as well as other bounds and other nutrients. However, it was not our intention to present a comprehensive diet model, but to demonstrate the impact of different achievement functions. The presented diet model uses a continuous decision variable Xi to denote intake of food i. The model could therefore propose a diet with, for example, 2 g apple (unrealistically low) or 17 g margarine and 13 g low-fat margarine (whereas a consumer would probably want to use either margarine or low-fat margarine but not both). Such issues can be overcome by extending the model with binary variables that indicate whether or not a food is actually used, see Supplementary Appendix D. This paper provides methodological insight into several GP achievement functions: MinSum, MinMax and EGP. It shows that the EGP achievement function is able to generate a solution that minimizes the sum of all violations, as well as a solution that minimizes the largest violation, and compromises between them. The EGP achievement function thus provides a way to obtain a range of solutions from one set of data and weights. CONFLICT OF INTEREST The authors declare no conflict of interest.
ACKNOWLEDGEMENTS This research received no grant from any funding agency in the public, commercial or not-for profit sectors.
REFERENCES 1 Buttriss JL, Briend A, Darmon N, Ferguson EL, Maillot M, Lluch A. Diet modelling: how it can inform the development of dietary recommendations and public health policy. Nutr Bull 2014; 39: 115–125. 2 Darmon N, Ferguson E, Briend A. Linear and nonlinear programming to optimize the nutrient density of a population's diet: an example based on diets of preschool children in rural Malawi. Am J Clin Nutr 2002; 75: 245–253. 3 Briend A, Darmon N, Ferguson E, Erhardt JG. Linear programming: a mathematical tool for analyzing and optimizing children's diets during the complementary feeding period. J Pediatr Gastroenterol Nutr 2003; 36: 12–22.
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7 4 Thompson S, Gower R, Darmon N. LiveWell for Life. A balance of healthy and sustainable food choices for France, Spain, and Sweden 2013: http://livewellforlife. eu/livewell-plate/reports accessed 25 April 2014 WWF Report. 5 Anderson AM, Earle MD. Diet planning in the third world by linear and goal programming. J Operation Res Soc 1983; 34: 9–16. 6 Ferguson EL, Darmon N, Fahmida U, Fitriyanti S, Harper TB, Premachandra IM. Design of optimal food-based complementary feeding recommendations and identification of key "problem nutrients" using goal programming. J Nutr 2006; 136: 2399–2404. 7 Fletcher LR, Soden PM, Zinober ASI. Linear programming techniques for the construction of palatable human diets. J Operation Res Soc 1994; 45: 489–496. 8 Darmon N, Ferguson EL, Briend A. A cost constraint alone has adverse effects on food selection and nutrient density: an analysis of human diets by linear programming. J Nutr 2002; 132: 3764–3771. 9 Darmon N, Ferguson EL, Briend A. Impact of a cost constraint on nutritionally adequate food choices for French women: an analysis by linear programming. J Nutr Educ Behav 2006; 38: 82–90. 10 Maillot M, Vieux F, Amiot MJ, Darmon N. Individual diet modeling translates nutrient recommendations into realistic and individual-specific food choices. Am J Clin Nutr 2010; 91: 421–430. 11 Masset G, Monsivais P, Maillot M, Darmon N, Drewnowski A. Diet optimization methods can help translate dietary guidelines into a cancer prevention food plan. J Nutr 2009; 139: 1541–1548. 12 Jones D, Tamiz M. Practical Goal Programming. Springer Science+Business Media, LLC: Boston, MA, 2010. 13 Tamiz M, Jones D, Romero C. Goal programming for decision making: an overview of the current state-of-the-art. Eur J Operation Res 1998; 111: 569–581. 14 Romero C. Extended lexicographic goal programming: a unifying approach. Omega 2001; 29: 63–71. 15 Romero C. A general structure of achievement function for a goal programming model. Eur J Operation Res 2004; 153: 675–686. 16 Jones D. A practical weight sensitivity algorithm for goal and multiple objective programming. Eur J Operation Res 2011; 213: 238–245. 17 Brill ED Jr. The use of optimization models in public-sector planning. Manag Sci 1979; 25: 413–422. 18 Makowski D, Hendrix EMT, Van Ittersum MK, Rossing WAH. A framework to study nearly optimal solutions of linear programming models developed for agricultural land use exploration. Ecol Model 2000; 131: 65–77.
19 Makowski D, Hendrix EMT, Van Ittersum MK, Rossing WAH. Generation and presentation of nearly optimal solutions for mixed-integer linear programming, applied to a case in farming system design. Eur J Operation Res 2001; 132: 425–438. 20 Romero C, Tamiz M, Jones DF. Goal programming, compromise programming and reference point method formulations: linkages and utility interpretations. J Operation Res Soc 1998; 49: 986–991. 21 Claassen GDH, Hendriks THB, Hendrix EMT. Decision Science: Theory and Applications. Wageningen Academic Publishers: Wageningen, 2007. 22 Gedrich K, Hensel A, Binder I, Karg G. How optimal are computer-calculated optimal diets? Eur J Clin Nutr 1999; 53: 309–318. 23 Wirsam B, Hahn A, Uthus EO, Leitzmann C. Fuzzy sets and fuzzy decision making in nutrition. Eur J Clin Nutr 1997; 51: 286–296. 24 Wirsam B, Uthus EO. The use of fuzzy logic in nutrition. J Nutr 1996; 126: 2337S–2341S. 25 Yaghoobi MA, Tamiz M. A method for solving fuzzy goal programming problems based on MINMAX approach. Eur J Operation Res 2007; 177: 1580–1590. 26 Dantzig GB. On the significance of solving linear programming problems with some integer variables. Econometrica 1960; 28: 30–44. 27 Nordic council of Ministers. Nordic Recommendations 2012. Integrating Nutrition and Physical Activity, 5th edn. Copenhagen: Nordisk Ministerråd, 2014. 28 Health Council of the Netherlands. Guidelines for a Healthy Diet 2006. Health Council of the Netherlands: The Hague, 2006. 29 Jones D, Jimenez M. Incorporating additional meta-objectives into the extended lexicographic goal programming framework. Eur J Operation Res 2013; 227: 343–349. 30 Carlson A, Lino M, Fungwe T. The Low-Cost, Moderate-Cost, and Liberal Food Plans. http://www.cnpp.usda.gov/Publications/FoodPlans/MiscPubs/Food Plans2007AdminReport.pdf. 2007 accessed on 25 April 2014: CNPP-20. 31 Gao X, Wilde PE, Lichtenstein AH, Tucker KL. The 2005 USDA food guide pyramid is associated with more adequate nutrient intakes within energy constraints than the 1992 pyramid. J Nutr 2006; 136: 1341–1346. 32 Cleveland LE, Escobar AJ, Lutz SM, Welsh SO. Method for identifying differences between existing food intake patterns and patterns that meet nutrition recommendations. J Am Diet Assoc 1993; 93: 556–563. 33 Bazaraa MS, Sherali HD, Shetty CM. Nonlinear Programming: Theory and Algorithms. Wiley: New York, 1993.
Supplementary Information accompanies this paper on European Journal of Clinical Nutrition website (http://www.nature.com/ejcn)
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European Journal of Clinical Nutrition (2015) 1 – 7
ORIGINAL ARTICLE
Diet models with linear goal programming: impact of achievement functions JC Gerdessen1 and JHM de Vries2 BACKGROUND/OBJECTIVES: Diet models based on goal programming (GP) are valuable tools in designing diets that comply with nutritional, palatability and cost constraints. Results derived from GP models are usually very sensitive to the type of achievement function that is chosen. This paper aims to provide a methodological insight into several achievement functions. It describes the extended GP (EGP) achievement function, which enables the decision maker to use either a MinSum achievement function (which minimizes the sum of the unwanted deviations) or a MinMax achievement function (which minimizes the largest unwanted deviation), or a compromise between both. An additional advantage of EGP models is that from one set of data and weights multiple solutions can be obtained. SUBJECTS/METHODS: We use small numerical examples to illustrate the ‘mechanics’ of achievement functions. Then, the EGP achievement function is demonstrated on a diet problem with 144 foods, 19 nutrients and several types of palatability constraints, in which the nutritional constraints are modeled with fuzzy sets. RESULTS: Choice of achievement function affects the results of diet models. CONCLUSIONS: MinSum achievement functions can give rise to solutions that are sensitive to weight changes, and that pile all unwanted deviations on a limited number of nutritional constraints. MinMax achievement functions spread the unwanted deviations as evenly as possible, but may create many (small) deviations. EGP comprises both types of achievement functions, as well as compromises between them. It can thus, from one data set, find a range of solutions with various properties. European Journal of Clinical Nutrition advance online publication, 22 April 2015; doi:10.1038/ejcn.2015.56 INTRODUCTION Mathematical modeling of diets can be defined as the use of mathematical techniques to formulate and optimize diets.1 We refer to Buttriss et al.1 for a description of relevance, history and applications of diet (planning) models based on linear programming (LP). LP-based diet models contain decision variables, an objective function and a set of constraints. Commonly, the decision variables are defined as: Xi = (proposed) daily intake of food i (i = 1...I), Xi ⩾ 0. The objective function minimizes (or maximizes) a linear function of the decision variables, for example total cost or total energy content of the diet. The (linear) constraints ensure that the proposed diet meets requirements on, for example, nutrient content and palatability. If the model contains integer variables, it is called a mixed integer LP model. Classification of diet models We distinguish between two classes: 1. Single-objective problems: nP minimize o (or maximize) one linear I function of Xi: Minimize i¼1 ci X i . If ci represents the cost of food i, then the objective function minimizes total diet cost and so on.2–4 Commonly, single-objective problems are formulated as straightforward (MI)LP models. 2. Multi-objective problems: two types are observed in literature: 2a. With the set of available foods no diet can be planned that complies with all constraints.5–7 A common approach is to search for a ‘best possible’ diet, which violates the
constraints as little as possible. This is a problem with multiple objectives: Minimize{Violation of constraint 1}, Minimize{Violation of constraint 2}, …, Minimize{Violation of the last constraint}. 2b. The decision maker aims to plan a diet that complies with all constraints, and that differs as little as possible from the actual diet (which does not comply with all constraints).4,8–11 In other words, (s)he wants to minimize the differences between the proposed (optimized) diet XP and the actual diet XA: Minimize X A1 - X P1 ; Minimize X A2 - X P2 ; :::; Minimize X AI - X PI :
This paper focuses on multi-objective diet models in Class 2a. For completeness, Class 2b is discussed in Supplementary Appendix A.
Multi-objective problems—searching Pareto-optimal solutions The objectives of a multi-objective problem are usually conflicting; commonly no solution exists that optimizes all objectives at the same time. For instance, the diet with the lowest violation of a constraint on iron intake might have a considerable violation of a constraint on intake of saturated fat, and vice versa. Methods for generating solutions for multi-objective problems therefore commonly focus on finding so-called Pareto-optimal solutions (also denoted as efficient solutions). Solution X is called Paretooptimal if no other solution X# exists that performs at least as good as X with respect to all objectives, and better with respect to at least one objective.12 In other words, the achieved value for one
1 Group Operations Research and Logistics, Wageningen University, Wageningen, The Netherlands and 2Division of Human Nutrition, Wageningen University, Wageningen, The Netherlands. Correspondence: JC Gerdessen, Operations Research and Logistics, Wageningen University, Hollandseweg 1, Wageningen 6706 KN, The Netherlands. E-mail: [email protected] Received 2 November 2014; revised 25 February 2015; accepted 26 February 2015
Achievement functions in diet models JC Gerdessen and JHM de Vries
2 objective cannot be improved without worsening the level of another objective.13 The concept of Pareto-optimality is illustrated in Table 1, which shows four fictitious diets, and their violations vFe and vSF of intake constraints on iron and saturated fat, respectively. No diet exists that has both lowest vFe and lowest vSF. We therefore aim to identify the Pareto-optimal diets. All references mentioned in Class 2 use linear goal programming (GP). GP uses the following steps12 to find ‘best possible’ (that is, Pareto-optimal) diets for problems in Class 2a: ● ●
(GP1): quantify the extent to which a diet violates the constraints. (GP2): minimize a function of these violations in order to obtain a diet that violates the constraints as little as possible. This function is called the achievement function.
Multi-criteria decision-making literature, in which GP is positioned, recognizes achievement function selection, weight selection and weight space analysis as topics of major importance for the quality of decision making,13–16 because any choice made in formulating the achievement function uses judgment of the modeler and implies assumptions on the preference structure of the decision maker. If the election of the achievement function is wrong, then the decision maker will probably not accept the solution.15 Results derived from GP models usually are very sensitive to the type of achievement function that is chosen.15 All references mentioned in Class 2 use a weighted additive achievement function. GP literature offers other achievement functions as well.15 Typically, with a weighted additive achievement function, each set of data and weights results in a single solution. It would be useful to offer the decision maker more than just this single Table 1.
A B C D
MATERIALS AND METHODS Goal programming Numerical example. In order to reveal the ‘mechanics’ of achievement functions, we use a simplified diet model with two foods: (1) bread and (2) meat, with associated decision variables X1 (X2) = amount of bread (meat) in the diet. An upper bound on salt intake restricts bread consumption to three units or less (nutritional constraint, Eq. (1)). An upper bound on saturated fat intake restricts meat consumption to two units or less (nutritional constraint, Eq. (2)). For sufficient iron intake the diet should contain at least six units of bread and/or meat (nutritional constraint, Eq. (3)): X1 X2 X1 þ X2 X1; X2 ⩾ 0
ð1Þ ð2Þ ð3Þ
Iron: vFe
Saturated fat: vSF
0 4 6 3
11 3 4 8
Pareto-optimal or not?
Yes Yes No: B has lower vFe and vSF Yes
3 ð1Þ 2 ð2Þ 6 ð3Þ
⩽ 3 ⩽ 2 ⩾ 6
X1 X2 þ X2
X1
- X1 - X1
þd1þd2þd3-
X2 X2
þ
- dþ 1 - dþ 2 - dþ 3
¼ 3 ¼ 2 ¼ 6
For instance, (X1,X2) = (1,1) implies d1- ¼ 2; d 2- ¼ 1; d 3- ¼ 4 and (X1,X2) = (6,1) þ implies d þ 1 ¼ 3; d 2 ¼ 1; d 3 ¼ 1. (Note that in any optimal solution either one of the deviational variables (d–j or d+j ) equals zero, or both.12 ) As initial constraint (3) provides a lower bound for X1+X2, it is allowed to have a positive deviation d þ 3 . Any shortage d 3 , however, violates nutritional constraint (3) and is therefore considered an unwanted deviation. Likewise, þ dþ 1 and d 2 are unwanted deviations, see Figure 1b. Next, step (GP2) searches the ‘best possible’ diet by formulating and minimizing the achievement function.
b 4
X2 →
X2 →
a
⩽ ⩽ ⩾
Figure 1a shows that constraints (1)–(3), are conflicting: no diet (X1, X2) exists that complies with all constraints. The resulting model belongs to Class 2a. Step (GP1) quantifies the violation of the nutritional constraints by adding deviational variables dj- ; dþ j ⩾0, which represent the negative and positive deviation from the right-hand side value of nutritional constraint j:
Example of Pareto-optimality Violation of intake constraint on
Diet Diet Diet Diet
solution and present a range of Pareto-optimal solutions. Offering multiple solutions allows choice of a solution which is most suitable for a specific decision problem, and that best meets nonquantifiable goals and preferences.17–19 This paper aims to provide methodological insight into several GP achievement functions: MinSum, MinMax and extended GP (EGP). It shows that the EGP achievement function is able to generate a solution that minimizes the sum of all violations, as well as a solution that minimizes the largest violation, and compromises between them. The EGP achievement function thus provides a way to obtain a range of solutions from one set of data and weights.
4
3
3
d2+ 2
2
––
6 X1 →
–
5
d3
4
6
3
+
+=
X1+ d1––d1+= 3 0
d3
d3
2
d1+
d3
1
+
d1–
X1≤ 3 0 0
(6,1)
2
(1,1) 1
6
1
d2–
+X X1
≥ 2 +X X1
X2≤ 2
X2+ d2––d2+= 2
0
1
2
3
4
5
6 X1 →
Figure 1. (a) A diet model in Class 2a. No diet exists that complies with all constraints. (b) A GP model uses deviational variables to quantify þ the deviations from the targets. Deviations d 1- ; d 2- ; d þ ) are allowed by the original constraints. The dþ 3 ( 1 ; d 2 ; d 3 ( → ) are unwanted deviations. A MinSum GP model will propose a corner point (•) of the shaded area, no matter which set of weights is used. A MinMax GP model is able to propose solutions inside the shaded area, e.g., if all weights are equal then (X1,X2) = (31/3, 21/3) (m) is proposed (constraints, Eqs. (14)–(16), cannot be visualized in two dimensions. Therefore it cannot visualized in b that point (X1, X2) = (31/3, 21/3) is a corner point in the nine-dimensional space (X1, X2, Dsum, d–1, d+1 , d–2, d+2 , d–3, d+3 ). European Journal of Clinical Nutrition (2015) 1 – 7
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Achievement functions in diet models JC Gerdessen and JHM de Vries
3 MinSum achievement function. A weighted additive achievement function —which we refer to as MinSum achievement function—minimizes a weighted sum of the unwanted deviations. For model (1)–(3), the þ unwanted deviations are d þ 1 ; d 2 ; d 3 , which results in the following MinSum GP model:
●
●
þ þ þ - Minimize Dsum ¼ w þ 1 d 1 þ w 2 d 2 þ w 3 d 3 ð4Þ þ þ d1 - d1 ¼ 3 ð5Þ X1 X 2 þ d2- - dþ ¼ 2 ð6Þ 2 X 1 þ X 2 þ d3- - dþ ¼ 6 ð7Þ 3 þ þ ð8Þ X 1 ; X 2 ; Dsum ; d 1- ; dþ 1 ; d2 ; d2 ; d3 ; d3 ⩾ 0 þ in which w þ 1 ; w 2 ; w 3 are user-defined, non-negative weights. The optimal solution of problem, Eqs. (4–8), depends on the weights. Independent of the weights, it is always located in the shaded area of Figure 1b. For þ wþ 1 ¼ w 2 ¼ w 3 all solutions in the shaded area have the same value of Dsum. As in LP-problems the optimal solution is always a corner point,12 an LP-solver will generate a corner point of the shaded area: (3,2) or (4,2) þ or (3,3). However, if the user specifies w þ 1 ¼ w 2 ¼ w 3 (s)he expresses that (s)he does not want to prioritize one nutrient, that is, (s)he cannot justify assigning one deviation more importance than another. Hence, it would be natural to obtain a balanced solution that spreads theþ total unwanted deviation over all three deviational variables dþ 1 ; d2 ; d3 instead of an unbalanced solution that piles the unwanted deviation on only one of them. Table 2 shows that the MinSum model is sensitive to weight changes: slight weight changes make the optimal diet ‘jump’ from one corner point to another. However, if the weights expressed by the user are similar, one would expect similar solutions. These imbalance and sensitivity are typical for an additive achievement function.12,14 Using an additive achievement function implies the assumption that all weighted unwanted deviations are additive and that nutritional adequacy of a diet is determined by the sum of its weighted unwanted deviations. This presupposes—implicitly—that
Table 2.
●
deviations can compensate each other, for example, an increase in the deviation from a vitamin C target can be compensated by a decrease in the deviation from a calcium target; tradeoffs between the deviations are precisely known, for example, 10% deviation from a vitamin C target is considered equally ‘grave’ as 5% deviation from a calcium target; and tradeoffs are constant and do not depend on intake level, for example, 10% extra deviation from a vitamin C target is considered equally ‘grave’ as 5% extra deviation from a calcium target, no matter whether the vitamin C intake is almost adequate or dangerously low.
MinMax achievement function. A MinMax achievement function (also called Chebyshev achievement function) aims to minimize the largest among the weighted unwanted deviations:14,15,20 þ þ þ - Minimize Dmax ¼ max w þ ð9Þ 1 d1 ; w 2 d2 ; w 3 d3 In order to obtain a linear model that minimizes this non-linear þ achievement function a constraint Dmax ⩾ w j- dj- ðDmax ⩾ w þ j d j Þ is added 21 for every unwanted deviation dj- ðdþ j Þ Minimize fDmax g þ d1- - dþ ¼ 3 X1 1 X 2 þ d2- - dþ ¼ 2 2 X 1 þ X 2 þ d3- - dþ ¼ 6 3 þ Dmax ⩾ w þ 1 d1 þ Dmax ⩾ w þ 2 d2 Dmax ⩾ w 3- d 3þ þ X 1 ; X 2 ; Dmax ; d 1- ; dþ 1 ; d2 ; d2 ; d3 ; d3 ⩾ 0
ð10Þ ð11Þ ð12Þ ð13Þ ð14Þ ð15Þ ð16Þ ð17Þ
Achievement function (10) and constraints together ensure that (14–16) þ þ þ - Dmax will take the value Dmax ¼ max w þ 1 d 1 ; w 2 d 2 ; w 3 d 3 , and that Dmax is as low as possible. Table 2 shows that using the MinMax achievement function for þ wþ 1 ¼ w 2 ¼ w 3 ¼ 1 results in a balanced solution: all unwanted
Solutions of the MinSum, MinMax and extended GP model Optimal solution of MinSum GP model (4)–(8) for various choices of the weights
Weights þ wþ 1 ; w2 ; w3 (1, 1, 1) (0.9, 1, 1) (1, 0.9, 1) (1, 1, 0.9)
Diet (X1,X2) (4,2) or (3,3) or (3,2) (4, 2) (3, 3) (3, 2)
þ Deviations dþ 1 ; d2 ; d3
deviations Weighted þ þ þ - wþ 1 d1 ; w2 d2 ; w 3 d3
Achievement function þ þ þ - Dsum ¼ w þ 1 d1 þ w2 d2 þ w 3 d3
(1,0,0) or (0,1,0) or (0,0,1)
(1,0,0) or (0,1,0) or (0,0,1)
1
(1, 0, 0) (0, 1, 0) (0, 0, 1)
(0.9, 0, 0) (0, 0.9, 0) (0, 0, 0.9)
0.9 0.9 0.9
Optimal solution of MinMax GP model (10)–(17) for various choices of the weights Weights þ wþ 1 ; w2 ; w3 (1, 1, 1) (0.9, 1, 1) (1, 0.9, 1) (1, 1, 0.9)
Diet (X1,X2) (3.333, (3.357, (3.321, (3.321,
2.333) 2.321) 2.357) 2.321)
þ Deviations dþ 1 ; d2 ; d3 (0.333, (0.357, (0.321, (0.321,
0.333, 0.321, 0.357, 0.321,
0.333) 0.321) 0.321) 0.357)
Achievement function þ þ þ - Dmax ¼ max w þ 1 d1 ; w 2 d2 ; w3 d3
deviations Weighted þ þ þ - wþ 1 d1 ; w 2 d2 ; w 3 d3 (0.333, (0.321, (0.321, (0.321,
0.333, 0.321, 0.321, 0.321,
0.333) 0.321) 0.321) 0.321)
0.333 0.321 0.321 0.321
Optimal solution of extended GP þmodel (11)–(18) for various choices of parameter λ. Weights w þ 1 ; w 2 ; w 3 ¼ ð0:5; 0:75; 1Þ were used. Parameter λ
Diet (X1,X2)
0.00 0.25 0.50 0.75 1.00
(4, (4, (3.60, (3.46, (3.46,
2) 2) 2.40) 2.31) 2.31)
þ Deviations d þ 1 ; d2 ; d3 (1, 0, 0) (1, 0, 0) (0.60, 0.40, 0) (0.46, 0.31, 0.23) (0.46, 0.31, 0.23)
deviations Weighted þ þ þ - wþ 1 d1 ; w 2 d2 ; w 3 d3
Dsum
Dmax
Dext = (1 − λ)Dsum+λ Dmax
(0.50, 0, 0) (0.50, 0, 0) (0.30, 0.30, 0) (0.23, 0.23, 0.23) (0.23, 0.23, 0.23)
0.50 0.50 0.60 0.69 0.69
0.50 0.50 0.30 0.23 0.23
0.50 0.50 0.45 0.35 0.23
Abbreviation: GP, goal programming.
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European Journal of Clinical Nutrition (2015) 1 – 7
Achievement functions in diet models JC Gerdessen and JHM de Vries
4 deviations are equal. Moreover, the model is less sensitive to weight changes: slight weight changes cause minor shifts in (X1,X2). The MinMax achievement function attempts to spread the unwanted deviations as evenly as possible and it enables to identify non-corner point solutions in the shaded area. However, possibly many unwanted deviations are found.14 With a MinMax achievement function the nutritional adequacy of a diet is mainly determined by its poorest nutrient. Extended GP achievement function. The MinSum and MinMax achievement functions can be combined into the so-called EGP achievement function Dext:14,15 Minimize fDext ¼ ð1 - λÞUDsum þ λUDmax g
ð18Þ
where parameter λ ∈ (0;1) weighs the importance attached to minimization of Dsum and Dmax. Dext comprises both Dsum and Dmax: using λ = 0 implies Dext = Dsum, so the model minimizes the weighted sum of the unwanted deviations. Using λ = 1 implies Dext = Dmax, so the model spreads the deviations and keeps the largest unwanted deviation as low as possible. For intermediate values of λ solutions are found that are compromises between the MinSum and MinMax solution. Thus, by varying λ the decision maker can obtain a set of dietary suggestions ‘between’ the MinSum and MinMax diets, see also Table 2. Normalization to (0;1)-interval; interpretation as fuzzy sets. In order to deal with issues of scaling and incommensurability it is useful to formulate a GP model in such a way that unwanted deviational variables are automatically normalized to a (0;1) interval.13 For instance, consider a diet model with two foods and two nutrients, see Table 3.
Definition dl j- ; dl þ j deviations from the left bound of target intake for nutrient j: dr j- ; dr þ j deviations from the right bound of target intake for nutrient j:
The dl j- and dr þ j are the unwanted deviations. The nutritional constraints can be formulated as 500X 1
þ
800X 2
þ
500X 1
þ
800X 2
þ
0:3X 1
þ
0:1X 2
þ
0:3X 1
þ
0:1X 2
þ
500dl 1dl 14000dr 1-
-
500dl þ 1
-
0:3dl 2dl 22:6dr 2-
-
4000dr þ 1 dr þ 1 0:3dl þ 2
-
2:6dr þ 2 dr þ 2
¼ 1500 ð19Þ ⩽ 1 ð20Þ ¼ 3000 ð21Þ ⩽ 1 ð22Þ ¼ 0:9 ð23Þ ⩽ 1 ð24Þ ¼ 2:4 ð25Þ ⩽ 1 ð26Þ
Thus, all unwanted deviational variables are normalized to [0;1], which facilitates judgment of tradeoffs. Moreover, the normalized deviational variables can be used to incorporate fuzzy sets for nutrient intake. Figure 2 shows a graph of intake I1 of nutrient 1 versus – + μ1 ¼ 1 - dl 1- - dr þ 1 . (In an optimal solution dlj and drj can never be both non-zero at the same time, so μ1 ðI1 Þ ¼ min 1 - dl 1- ; 1 - dr þ 1 ¼ 1 - dl 1- - dr þ 1 .) For 1500 ⩽ I1 ⩽ 3000 the intake of nutrient 1 is considered þ fully adequate: both unwanted deviational variables dl 1 and dr 1 are zero and μ1 ¼ 1: Intake I1 = 1000 is considered fully inadequate: dl 1- ¼ 1 and μ1 = 0. Likewise, I1 = 7000 has dr þ 1 ¼ 1 and μ1 = 0. So an adequate intake has μ1 = 1 and an inadequate intake has μ1 = 0. If we assume the adequacy of intake I1 increases linearly from 0 to 1 in interval (1000;1500), we can use μ1 as proxy for the adequacy of the diet with respect to nutrient 1. In case overall nutritional quality Μ of the diet is determined by the adequacy of its poorest nutrient we can calculate it as
Table 3.
þ M ¼ Minimumfμ1 ; μ2 g ¼ 1 - Maximum dl 1- ; dr þ ¼ 1 - Dmax , 1 ; dl 2 ; dr 2 which implies the MinMax achievement function yields diets with maximal M. The adequacy curve in Figure 2 can be interpreted as fuzzy set for the adequacy of intake I1 with membership function μ1 ðI1 Þ ¼ 1 - dl 1- - dr þ 1 . For more information on use of fuzzy sets for modeling intake adequacy we refer to Gedrich et al.,22 Wirsam et al.,23 Wirsam and Uthus.24 More information on LP-formulations for fuzzy sets is found in Yaghoobi and Tamiz.25 Mixed Integer LP-formulations of Dantzig26 can be used to construct curves with more than three intervals.
RESULTS The EGP achievement function is used to obtain ‘best possible’ solutions for a diet model for planning of diets for dietary controlled trials for men aged 19–30y. Palatability constraints define lower and upper bounds on intake of various foods (‘at most 245 g of bread’), and they link intakes of foods (‘3–7 g spread per slice’). Nutritional constraints are formulated via adequacy curves for 19 nutrients and for vegetables and fruits. The four characteristic points of each adequacy curve are defined in the following way: a is the Lower Intake Level below which an intake could lead to risk in most individuals; b is the average requirement, sufficient for virtually 50% of healthy people in a group, c is the recommended daily intake which is sufficient for nearly all people; d is the upper intake level that is unlikely to pose a risk of adverse health effects.27 If no information was available a nutrition expert (JdV) made an estimate.27,28 Energy intake was fixed to 100% of the estimated average requirements. Supplementary Appendix B provides the full model. The model was programmed in Fico Xpress 7.0.1 (Fair Isaac Corporation, London, UK) and calculations were done on an HP desktop (Hewlett-Packard Company, Palo Alto, CA, USA) with Intel I7 processor. The model generates the 11 diets in one second. Table 4 provides reference values for the adequacy curves and summarizes results for λ ∈ {0;0.1;…;1}. Using λ = 0 implies Dext = Dsum, so the model finds the diet with minimal sum of unwanted deviations. In this MinSum diet all nutrient intakes except mono- and disaccharides (MDSacch) are in their optimal range (at plateau [b,c] of the adequacy curve). However, this diet has dr þ MDSacch ¼ ð7:30 - 5Þ=ð10 - 5Þ ¼ 0:460, which means that MDSacch intake is far too high and has a suboptimal adequacy μMDSacch = 1 − 0.460 = 0.540. As MDSacch is the only nutrient with
Figure 2.
Adequacy curve for nutrient 1.
Data for diet model Nutrient content
Nutrient 1 Nutrient 2
Nutritional constraints
(mg in 100 g of food 1)
(mg in 100 g of food 2)
Lower bound on intake (mg)
Target intake (mg)
Upper bound on intake (mg)
500 0.3
800 0.1
1000 0.6
(1500; 3000) (0.9; 2.4)
7000 5
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© 2015 Macmillan Publishers Limited
en% en% en% en% en% en% mg mg en% gr/MJ mg mg mg mg mg mg μg mg μg g g
Protein Total fat SFA MUFA PUFA Linol EPA+DHA Cholesterol MDSacch Fiber Ca Fe K Vit B1 Vit B2 Vit B6 Vit B12 Vit C Folate Vegetables Fruits
8% 2% 0% 8% 0% 0% 350 0 0% 0 400 7 1600 0.5 0.8 0.8 1 10 100 150 100
a
10% 25% 0% 10% 3% 2% 450 0 0% 3 500 9 3100 0.9 1.1 1 1.4 50 200 200 200
b 20% 40% 10% 20% 10% 9% 3000 200 5% —b 800 23 3500 1.4 1.7 1.5 2 75 400 400c 300c
c 25% 45% 15% 28% 12% 10% 4000 300 10% —b 2500 60 10 000 7 8.5 25 10 375 2000 400c 300c
d 14.0% 40.0% 10.0% 17.6% 10.0% 8.82% 450 74.6 7.30% 3.00 720 12.1 3227 1.35 1.1 1.5 2.0 75 292 200 200 Dsum Dmax Dext
I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
dl − 0 0 0 0 0 0 0 0 0.460 0 0 0 0 0 0 0 0 0 0 0 0
dr+ 1 1 1 1 1 1 1 1 0.540 1 1 1 1 1 1 1 1 1 1 1 1 0.460 0.460 0.460
μ 1 1 1 1 1 1 1 1 0.691 1 1 1 1 1 1 1 1 0.846 1 1 1 0.463 0.309 0.448
0.1
1 1 1 1 1 1 1 1 0.692 1 1 1 1 1 1 1 0.990 0.854 1 1 1 0.463 0.308 0.432
0.2
1 1 1 1 1 1 1 1 0.700 1 1 1 1 1 1 1 0.981 0.852 1 1 1 0.466 0.300 0.416
0.3
1 1 1 1 1 1 1 1 0.702 1 1 1 1 1 1 1 0.981 0.850 1 1 1 0.468 0.298 0.400
0.4
1 1 1 1 1 1 1 1 0.703 1 1 1 1 1 1 1 0.980 0.848 1 1 1 0.469 0.297 0.383
μ
0.5
1 1 1 1 1 1 1 1 0.704 1 1 1 1 1 1 1 0.980 0.847 1 1 1 0.469 0.296 0.365
0.6
Results for λ = 0, 0.1, …, 1
1 1 1 1 1 1 1 1 0.729 1 1 1 1 1 1 0.993 0.926 0.835 1 1 1 0.518 0.271 0.345
0.7
1 1 1 1 1 1 1 1 0.782 1 1 1 1 1 1 0.993 0.911 0.866 1 1 0.782 0.666 0.218 0.307
0.8
1 1 1 1 1 1 1 1 0.820 0.820 1 1 1 0.856 1 0.979 0.869 0.867 1 1 0.820 0.969 0.180 0.259
0.9
19.2% 40.8% 10.8% 16.9% 10.3% 8.42% 434 180 5.82% 2.51 949 12.4 3745 2.32 1.26 2.15 3.31 124 309 192 184
Intake
0 0 0 0 0 0 0.163 0 0 0.163 0 0 0 0 0 0 0 0 0 0.163 0.163
dl −
1
0 0.163 0.163 0 0.163 0 0 0 0.163 0 0.088 0 0.038 0.163 0 0.028 0.163 0.163 0 0 0
dr+
1 0.837 0.837 1 0.837 1 0.837 1 0.837 0.837 0.912 1 0.962 0.837 1 0.972 0.837 0.837 1 0.837 0.837 1.95 0.163 0.163
μ
Abbreviations: EPA, eicosapentaenoic acid; DHA, docosahexaenoic acid; MDSacch, mono- and disaccharides, MUFA, mono-unsaturated fatty acid; PUFA, poly-unsaturated fatty acid; SFA, saturated fatty acid. a These units apply to a, b, c, d, I. The dl−, dr+, μ are dimensionless. bNo c and d are used, because within the feasible diets fiber intake will never be too high. cThese are palatability constraints.
unita
Nutrient
0
Reference values for nutritional adequacy curves and summary of results for λ = 0, 0.1, …, 1
Reference values for adequacy curves
Table 4.
Achievement functions in diet models JC Gerdessen and JHM de Vries
5
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Achievement functions in diet models JC Gerdessen and JHM de Vries
6 suboptimal intake the MinSum diet has Dmax = 0.460. Increasing λ implies that the model lowers Dmax at the cost of increasing Dsum, thus spreading the unwanted deviations in order to find a more balanced solution: for λ = 0.1 the unwanted deviation of MDSacch decreases to dr þ MDSacch ¼ 0:309 (so μMDSacch = 1 − 0.309 = 0.691) whereas vitamin C intake becomes suboptimal (μVitC = 0.846). For λ = 0.9 the largest deviation has decreased to 0.180. However, seven intakes are then suboptimal. For λ ¼ 1 the model strictly minimizes the largest unwanted deviation, which results in the diet with lowest Dmax. This MinMax diet has 14 suboptimal intakes and the highest Dsum. It is entirely up to the nutritionist to judge whether a decrease in Dmax is worth an increase in Dsum, and/or an increase in the number of violated constraints and to express a preference for any of the generated diets, based on the specific situation on hand. Supplementary Appendix C shows which foods were chosen. DISCUSSION This paper aims to provide a methodological insight into several GP achievement functions: MinSum, MinMax and EGP. A MinSum achievement function minimizes the sum of the unwanted deviations from nutritional targets and is thus appropriate in situations where diet quality is determined by the sum of these unwanted deviations. It can, however, lead to solutions that are unbalanced and that are sensitive to changes in preferential weights. A MinMax achievement function minimizes the largest among the unwanted deviations, and is thus appropriate when diet quality is mainly determined by the nutrient with the largest unwanted deviation. MinMax GP provides solutions that are as balanced as possible with respect to the unwanted deviational variables. However, possibly many unwanted deviations occur. An EGP achievement function is a compromise between MinSum and MinMax. It can—from one set of data and weights—obtain the MinSum solution, the MinMax solution, and a set of solutions ‘between’ the MinSum and MinMax solutions. Offering multiple solutions allows choice of a solution that is most suitable for a specific decision problem, and that best meets non-quantifiable goals and preferences.17–19 Extending a MinSum GP model to a MinMax GP model takes one additional variable (Dmax) plus one constraint for every unwanted deviation. In terms of model size (see also Supplementary Appendix B) this can be considered as very small. The EGP achievement function requires one extra model parameter (λ), which weighs the importance attached to minimizing the total unwanted deviation versus the importance of minimizing the largest unwanted deviation. No general rule can be given for setting the most useful value of λ. It seems most practical to let the model run for, for example, λ ∈ {0;0.1;…;1} and then judge the resulting diets and their nutritional adequacy. If desired, the model can be rerun with smaller step-sizes for λ in relevant sub-intervals, for example, λ ∈ {0.71;0.72;…;0.89}. In this way the decision maker is supported in finding his/her own tradeoffs. GP offers several other achievement functions that are worth exploring in diet modeling context. For instance, Lexicographic GP,13,15 in which the deviational variables are assigned to a number of priority levels that are minimized sequentially. In minimization runs for lower level deviational variables the higher level deviational variables are fixed to their (previously obtained) optimal values. Lexicographic GP requires the decision maker to provide a strict hierarchy of the unwanted deviations. Also, in GP it is possible to minimize the number of unmet nutritional constraints.29 This requires introducing binary variables indicating whether nutritional constraints have been met (see Supplementary Appendix D). It is useful in situations where unmet nutritional constraints incur costs, for example, due to necessary fortification programs. European Journal of Clinical Nutrition (2015) 1 – 7
A key consideration in achievement function selection is the preference structure of the decision maker.15 If the deviational variables can be classified into strict priority classes between which no finite tradeoffs exist then Lexicographic GP should be considered. If finite tradeoffs do exist between deviational variables then the decision maker should consider EGP, which offers the opportunity either to minimize the total deviation or the largest deviation or a compromise between both. A decision maker who wants to minimize the number of unmet nutritional constraints needs to introduce binary variables. It will depend on the type of nutrition question, for example whether it is aimed at the individual or population level, what the best modeling approach is. Further research is necessary to build a comprehensive framework that helps nutritionists to select the most suitable modeling approach for a wide range of diet problems. This paper focuses on linear achievement functions. In literature also models are described with quadratic achievement functions.22,30–32 In a quadratic achievement function the (weighted) sum of squared unwanted deviations is minimized, which means that large deviations are penalized more than small deviations.32 Quadratic achievement functions can find non-corner point solutions. However, possibly local optima are generated.33 The quality of the solutions of a diet model depends on the quality and choice of data.1 For our experiment, we chose a limited number of foods and nutrients, and had to base adequacy curves partly on expert opinion, because not all needed information was available in literature. Also, we did our experiment only for a specific diet of one energy level. Other foods could have been used, as well as other bounds and other nutrients. However, it was not our intention to present a comprehensive diet model, but to demonstrate the impact of different achievement functions. The presented diet model uses a continuous decision variable Xi to denote intake of food i. The model could therefore propose a diet with, for example, 2 g apple (unrealistically low) or 17 g margarine and 13 g low-fat margarine (whereas a consumer would probably want to use either margarine or low-fat margarine but not both). Such issues can be overcome by extending the model with binary variables that indicate whether or not a food is actually used, see Supplementary Appendix D. This paper provides methodological insight into several GP achievement functions: MinSum, MinMax and EGP. It shows that the EGP achievement function is able to generate a solution that minimizes the sum of all violations, as well as a solution that minimizes the largest violation, and compromises between them. The EGP achievement function thus provides a way to obtain a range of solutions from one set of data and weights. CONFLICT OF INTEREST The authors declare no conflict of interest.
ACKNOWLEDGEMENTS This research received no grant from any funding agency in the public, commercial or not-for profit sectors.
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Supplementary Information accompanies this paper on European Journal of Clinical Nutrition website (http://www.nature.com/ejcn)
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European Journal of Clinical Nutrition (2015) 1 – 7
