Diet models with linear Goal Programming: impact of achievement functions
10 February 2016
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Appendix A.
Model Class 2b: Minimize the differences between the actual
diet and the proposed diet
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Models in Class 2b search for diets that conform as closely as possible to an actual diet, while
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meeting constraints on e.g. nutrient recommendations, palatability and total cost.
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The composition of the actual and proposed diet can be expressed in amounts consumed per
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food group1;2 or in energy provided per food group3. These result in models of similar
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structure, and therefore we only elaborate models that express diets in amounts consumed per
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food group.
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Notation, GP constraint
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In this appendix we use the following notation:
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Qi
~
Data: Intake of food group i in actual diet (i = 1…I)
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Xi
~
Decision variable: Intake of food group i in proposed diet (i = 1…I)
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with I the number of food groups. The deviation between the actual diet and the proposed diet
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can be calculated via a GP constraint:
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X i  Qi d i  Qi d i  Qi
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d i , d i  0
for all food groups i
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Now di  di  di is the absolute value of the relative deviation between the actual and the
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proposed diet*. The negative and positive deviations d i , d i are normalized: di  1 means
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that the consumption in food group i of the proposed diet is 100% lower than in the actual
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diet, and di  1 means that the consumption in food group i of the proposed diet is 100%
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higher than in the actual diet.
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MinSum achievement function
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As an example we consider a diet problem with two foods. Actual food intakes are Q1 = 3 and
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Q2 = 2, and a nutritional constraint 2X1 + 3X2 ≥ 15 applies. The proposed diet should resemble
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the actual diet as much as possible. Therefore, all deviations d i , d i from the actual diet are
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regarded as unwanted. A MinSum GP model could look as follows:


Note that it is also possible to model absolute deviations: X i  d i  d i  Qi for all i. One should keep in
mind that any choice made in modeling the deviation uses judgment from the decision-maker and implies
assumptions on the underlying preference structure.
*
Diet models with linear Goal Programming: impact of achievement functions
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Minimize
D
sum
X1
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2 X1
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X2
 3X 2
 w1 d1  w1 d1  w2 d 2  w2 d 2
 3d1
 2d 2
 3d1
 2d 2

[A1]
 3 [A 2]
 2 [A3]
 15 [A 4]
X 1 , X 2 , Dsum , d1 , d1 , d 2 , d 2 , d3 , d3  0
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with w1 , w1 , w2 , w2 the non-negative weights assigned by the user.
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The shaded area in Figure A1 contains all diets that comply with nutritional constraint [A4].
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Of these we want to find the one that has minimal deviation from the actual diet. The set of
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Pareto-optimal solutions is indicated with a bold line segment connecting (3,3) and (4.5,2).
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Figure A1. Class 2b: The shaded area contains all diets that comply with nutritional constraint [A4]. Of
these we aim to find the diet with minimal deviation from the actual diet ( ). A MinSum GP model will
yield diet (4.5, 2) or diet (3,3) ( ). Diet (3.75, 2.5) ( ) is reachable via a MinMax GP model.
The optimal solution of model [A1]-[A4] depends upon the weights that are chosen. However,
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independent of the weights, it is always located within the Pareto set (i.e. the set of Pareto-
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optimal solutions). Note that all Pareto-optimal solutions have d1  d 2  0 . For w1  w2  1
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all Pareto-optimal solutions have total weighted deviation Dsum = 1. That means that –
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according to the MinSum GP model with w1  w2  1 – all Pareto-optimal diets are equally
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preferable. Due to the fact that in LP-problems the optimal solution is always found in a
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corner-point, an LP-solver will generate one of the corner-points of the efficient set: (4.5,2) or
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(3,3). However, if the user specifies that w1  w2 (s)he expresses that (s)he cannot justify
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assigning one deviation more importance than another, and therefore it would be natural to
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obtain a balanced solution that spreads the deviations over both deviational variables
Diet models with linear Goal Programming: impact of achievement functions
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( d1 , d 2 ) instead of an unbalanced solution that piles the unwanted deviation on only one of
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them.
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Table A1. Optimal solution of MinSum GP model [A1]-[A4] for various choices of the weights.
Optimal solution
Weights
Diet
Deviations
Weighted deviations
Achievement function
( w1 , w2 )
( X1 , X 2 )
( d1 , d 2 )
( w1 d1 , w2 d 2 )
(0.9, 1)
(4.5, 2)
(0.5, 0)
(0.45, 0)
0.45
(1, 1)
(4.5, 2) or (3, 3)†
(0.5, 0) or (0, 0.5)
(0.45, 0) or (0, 0.45)
0.45
(1, 0.9)
(3, 3)
(0, 0.5)
(0, 0.45)
0.45


Dsum = w1 d1
 w2 d 2
†
According to the MinSum achievement function all diets on line segment (4.5, 2) – (3, 3) are equally
preferable, because they all have the same achievement function value: 0.45. However, an LP-solver will only
generate a corner-point: (4.5, 2) or (3, 3).
Table A1 shows the optimal solution of model [A1]-[A4] for various choices of the weights
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w1 , w2 . It shows that the MinSum model is sensitive to weight changes: slight changes in the
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weights make the optimal diet ‘jump’ from one corner-point to another.
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MinMax achievement function
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The MinMax GP version of model [A1]-[A4] is
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Minimize { Dmax } [A5]
X1
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2 X1
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X2
 3X 2
 3d1
 2d 2
 3d1
 2d 2
 3 [A 2]
 2 [A3]
 15 [A 4]
Dmax  w1 d1
Dmax  w1 d1
[A6]
[A7]
Dmax  w2 d 2
Dmax  w2 d 2
[A8]
[A9]
X1, X2 ≥ 0; Dmax , d1 , d1 , d 2 , d 2  0
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Table A2 shows the optimal solution of model [A2]-[A9] for various choices of the weights
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w1 , w2 . Now, for w1  w2  1 a balanced solution is obtained, in which all unwanted
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deviations are equal. Moreover, the model is less sensitive to weight changes: slight changes
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in the weights cause minor shifts in the optimal solution. The MinMax GP model represents a
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situation in which the resemblance between two diets is determined by the food that differs
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the most.
Diet models with linear Goal Programming: impact of achievement functions
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Table A2. Optimal solution of model MinMax GP model [A2]-[A9] for various choices of the weights.
Optimal solution
Weights
Diet
Deviations
Weighted deviations
( w1 , w2 )
( X1 , X 2 )
( d1 , d 2 )
( w1 d1 , w2 d 2 )
Achievement function
Dmax
(0.9, 1)
(3.79, 2.47)
(0.263, 0.237)
(0.237, 0.237)
0.237
(1, 1)
(3.75, 2.50)
(0.250, 0.250)
(0.250, 0.250)
0.250
(1, 0.9)
(3.71, 2.53)
(0.237, 0.263)
(0.237, 0.237)
0.237
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Extended GP achievement function
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The achievement functions of the MinSum and MinMax GP model can be combined into the
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achievement function Dext of a so-called Extended GP model 4;5:
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Minimize
Dext
 (1  )  Dsum    Dmax 
[A10]
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where parameter   0;1 weighs the importance attached to the minimization of the MinSum
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and MinMax achievement function. An Extended GP model can find solutions that are
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compromises between the MinSum and MinMax solutions. It can thus provide valuable
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dietary suggestions that are located ‘between’ the MinSum and MinMax diets. However, the
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numerical example is too small to contain any of these intermediate solutions.
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