A Genetic Algorithm Approach to Optimize Planning of
Food Fortification
Department of Statistics, ISU
Dave Osthus
March 14, 2011
Dave Osthus (ISU)
A GA Optimization Approach
March 14, 2011
1 / 27
Outline
1
Malnutrition Background
2
Current Fortification Planning Approach
3
Data Description
4
Optimal Fortification Planning Approach
5
Genetic Algorithms
6
Results from the Implementation of the Genetic Algorithm
7
Future Work
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Malnutrition Background
Nutrition Definitions
Malnutrition: The inadequate, excessive, or unbalanced consumption
of nutrients.
Vitamin A: A vitamin composed of many forms that plays a role in
vision, immune function and skin health.
Retinol: Absorbed through animal sources.
Non-Retinol: Absorbed through vegetable sources.
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Malnutrition Background
Malnutrition Background Continued
Prevalence of vitamin A inadequacy: Proportion of individuals in a
population with vitamin A consumption below their requirement.
Estimated as the proportion of individuals in a population with
vitamin A consumption below the Estimated Average Requirement
(EAR).
EAR: Daily nutrient intake level that is estimated to meet the needs of
half the healthy individuals in a specified age and gender population.
Prevalence of retinol excess: Proportion of individuals in a
population with retinol consumption above their upper limit.
Estimated as the proportion of individuals in a population with retinol
consumption above the Tolerable Upper Limit (UL).
UL: Highest level of nutrient consumption regarded as safe for
individuals in a specified age and gender population.
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Malnutrition Background
Fortification Definitions
Food Fortification: An intervention that adds nutrients to food
sources.
For vitamin A food fortification, retinol is the fortificant.
Intervention: A method or approach to correct a nutrient deficiency.
Program: The totality of all interventions to combat a nutrient
deficiency.
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Malnutrition Background
Questions of Interest
How are populations/subpopulations identified as malnourished?
Given a population/subpopulation is identified as malnourished, how
is a food fortification plan selected?
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How are populations/subpopulations identified as malnourished?
Estimating Usual Daily Intake Distributions
Usual nutrient intake is the long-run average of daily nutrient intakes.
Nusser et. al. (1996) propose a method for estimating usual daily
intake distributions.
1
2
3
4
5
6
Requires multiple daily nutrient intake measurements from a subsample
of individuals.
Nuissance effects are removed, such as interview day of week, season of
year, or interview repetition.
Utilizes a semi-parametric transformation to transform daily nutrient
intakes in the original scale to daily nutrient intakes in the normal scale.
Fits a measurement error model to the daily nutrient intakes in the
normal scale.
Estimates moments for the usual daily intake distribution in the normal
scale.
Fits a grafted polynomial to transform usual daily intake distribution in
the normal scale to usual daily intake distribution in the original scale.
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Current Approach to Fortification Planning
Current Approach to Fortification Planning
Iterative approach to determine the effectiveness of a proposed
fortification plan (Carriquiry 2010).
1
2
3
Obtain baseline estimates of the prevalence of inadequate usual
nutrient intake.
Establish a goal for the intervention.
Determine fortification approach.
Select candidate food vehicles and fortification limits.
Simulate data as if proposed fortification planning strategy had been
implemented.
4
Estimate the usual nutrient intake distributions using the simulated
data. Observe prevalence of nutrient adequacy in target group as well
as all other groups. If these quantities are acceptable, then an effective
fortification strategy has been found.
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Ugandan Data
Data Example
Measurements collected by the International Food Policy Research
Institute
Ugandan children between 6 and 24 months of age.
Vitamin A and retinol are the nutrients of interest. Daily retinol
intakes calculated as 60% of daily vitamin A intakes.
Sugar, vegetable oil, wheat flour and maize flour are the food vehicles
of interest.
437 children in sample. Two days of measurements on 41 children.
Individual
Day
1
2
3
4
5
5
1
1
1
1
1
2
Vitamin A
(µg RAE)
169.40
128.78
22.88
324.40
192.91
168.93
Retinol
(µg)
101.64
77.27
13.73
194.64
115.75
101.36
Sugar
(g)
45.31
17.86
19.08
69.89
50.17
56.50
Vegetable Oil
(g)
2.17
1.39
1.10
1.54
8.09
6.65
Wheat Flour
(g)
0.00
18.10
0.00
0.00
43.74
25.46
Maize Flour
(g)
18.79
0.00
0.00
0.00
0.00
0.00
Table: Data collected on Ugandan children between 6 and 24 months of age.
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Ugandan Data
Summary of Data
Skewed right consumption for both nutrients and all food vehicles.
Daily Vitamin A Consumption for Ugandans
between 6 and 24 Months of Age
Daily Sugar Consumption for Ugandans
between 6 and 24 Months of Age
250
150
200
count
count
150
100
100
50
50
0
0
0
Figure:
500
1000
1500
Vitamin A Consumption µg RAE/day
2000
0
50
100
150
200
250
Sugar Consumption g/day
Distributions of daily vitamin A (µg RAE) and sugar (g) consumption.
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Ugandan Data
Notation
N : the number of people in our sample.
Y∗Aij : the observed, daily intake of vitamin A consumed prior to
nutrient fortification by person i, i ∈ {1, 2, . . . , N} on day j,
j ∈ {1, 2, . . . , J}.
Y∗Rij : the observed, daily intake of retinol consumed prior to nutrient
fortification by person i on day j.
U∗Ai : estimated usual vitamin A intake for person i, prior to nutrient
fortification.
U∗Ri : estimated usual retinol intake for person i, prior to nutrient
fortification.
Vijk : the dietary intake of food vehicle k, {k ∈ 1, 2, . . . , K }
consumed by person i on day j.
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Ugandan Data
Notation Continued
YAij : simulated, fortified daily vitamin A intake by person i on day j.
PK
∗ +
YAij = YAij
k=1 (αk ∗ Vijk )
YRij : simulated, fortified daily retinol intake by person i on day j.
PK
∗ +
YRij = YRij
k=1 (αk ∗ Vijk )
αk : the amount of retinol added to one unit of food vehicle k.
αk ∈ [0, fortification limit for food vehicle k]
UAi : estimated usual vitamin A intake for person i, post fortification.
URi : estimated usual retinol intake for person i, post fortification.
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Ugandan Data
Estimated Usual Vitamin A and Retinol Intake
Distributions
Usual Nutrient Intake of Vitamin A using ISU Method
Usual Nutrient Intake of Retinol using ISU Method
0.005
0.008
0.004
density
density
0.006
0.003
0.004
0.002
0.002
0.001
0.000
0.000
0
200
400
600
800
1000
0
Vitamin A Consumption (µg RAE/day)
200
400
600
800
1000
Retinol Consumption (µg/day)
Figure: Usual daily intake distributions of vitamin A and retinol estimated using the ISU
method. The vertical line for vitamin A is the EAR = 286 µg RAE/day. The vertical
line for retinol is the UL = 600 µg/day. The estimated prevalence of vitamin A
inadequacy is 92.84%. The estimated prevalence of retinol excess is 0%.
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Ugandan Data
Proposed Fortification Plan Example
Intervention Goals: Prevalence of vitamin A inadequacy of 50% and
prevalence of retinol excess of 15%.
Fortification Limits:
Sugar = 10.0 µg retinol/g
Vegetable Oil = 25.0 µg retinol/g
Wheat Flour = 2.4 µg retinol/g
Fortification Plan:
Sugar = 7.0 µg retinol/g
Vegetable Oil = 25.0 µg retinol/g
Wheat Flour = 2.0 µg retinol/g
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Ugandan Data
Fortification Plan Implementation Results
Estimated Usual Intake Distribution for Vitamin A
Estimated Usual Intake Distribution for Retinol
0.006
0.008
0.005
0.006
density
density
0.004
0.003
0.004
0.002
0.002
0.001
0.000
0.000
0
200
400
600
800
1000
1200
0
200
Vitamin A Consumption (µg RAE/day)
400
600
800
1000
1200
1400
Retinol Consumption (µg/day)
Figure: Usual daily intake distributions of vitamin A and retinol estimated using the ISU
method. The vertical line for vitamin A is the EAR = 286 µg RAE/day. The vertical line for
retinol is the UL = 600 µg/day. The estimated prevalence of vitamin A inadequacy is 47.51%.
The estimated prevalence of retinol excess is 13.36%.
Dave Osthus (ISU)
A GA Optimization Approach
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Ugandan Data
Analysis of Current Fortification Planning Approach
Weaknesses to Current Fortification Planning Approach
Guess and check methods can be time consuming.
Difficulties arise when a large number of food vehicles are considered.
No consideration of cost.
Goal for a Fortification Planning Approach
Meet the goals of the intervention for minimal cost.
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Optimal Fortification Planning Approach
Optimization Function
We wish to minimize a function that satisfies two criteria:
1
Achieves target prevalence (Constraint)
2
Minimizes cost (Cost)
The function will be of this form:
f (α1 , α2 , . . . , αK , λ) = g (α1 , α2 , . . . , αK ) + λ(h(α1 , α2 , . . . , αK |δEAR , δUL ) − d)
where
g (α1 , α2 , . . . , αK ) =
K
X
ck ∗ αk ,
ck ≥ 0
k=1
and
h(α1 , α2 , . . . , αK |δEAR , δUL ) − d =(|Estimated prevalence of vitamin A adequacy − δEAR |
+ |Estimated prevalence of retinol adequacy − δUL |) − 0.
where δEAR and δUL are the target prevalences for vitamin A and retinol adequacy, respectively.
Dave Osthus (ISU)
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Optimal Fortification Planning Approach
Optimization Function Continued
Use the method of Lagrangian multipliers to optimize
f (α1 , α2 , . . . , αK , λ). Thus take
∂
f (α1 , α2 , . . . , αK , λ)=0,
ˆ
∀k
∂αk
∂
f (α1 , α2 , . . . , αK , λ)=0.
ˆ
∂λ
Problem:
∂
f (α1 , α2 , . . . , αK , λ)
∂αk
is not analytically tractable due to how “Estimated prevalence of
nutrient adequacy” is estimated for both vitamin A and retinol.
A numerical optimization approach will be utilized.
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Genetic Algorithm
Genetic Algorithm
A genetic algorithm (GA) is a stochastic optimization algorithm
that attempts to mimic the evolutionary process as demonstrated in
nature by biological individuals.
GA’s are inspired by Charles Darwin’s theory of evolution.
1
2
3
Within a biological population, there is variability amongst the
individuals, much of which can be inherited.
Some individuals in a biological population possess certain
characteristics making them more likely to reproduce and leave their
genetic makeup to future generations. Thus, some individuals are more
fit than others.
Over the course of many generations, individuals will possess the
characteristics that allowed their ancestors to reproduce, while the
traits that made individuals of previous generations less likely to
reproduce will dissipate from the biological population.
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Genetic Algorithm
Paralleling Terminology
Evolutionary Terms
Biological population
Mathematical Terms
Function domain
Biological individual
Input ∈
function domain
Mathematical function
How characteristics
relate to fitness
Fitness
Dave Osthus (ISU)
Output of
mathematical function
Nutrition Terms
[0, fortification limit for food vehicle 1] x
[0, fortification limit for food vehicle 2] x
.
.
.
[0, fortification limit for food vehicle K]
(α1 , α2 , . . . , αK )
f (α1 , α2 , . . . , αK , λ)=Z
ˆ
Z is the fitness of (α1 , α2 , . . . , αK )
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Genetic Algorithm
Genetic Algorithm Process
The GA process can be broken down into seven steps.
1
Generate initial genetic population
A collection of candidate solutions, (α1 , α2 , . . . , αK )s.
2
Evaluate fitness of initial genetic population
Calculate the Zs for each (α1 , α2 , . . . , αK ).
3
Selection phase
Select a subset of candidate solutions from the initial genetic population with more
optimal fitness, Z, than the unselected subset.
4
Recombination phase
Pair two candidate solutions, each called a parent.
Exchange the appropriate genes (segments of the chromosome) between the
chromosomes, αk s, of the paired parents.
This produces two new candidate solutions, or offspring.
5
Mutation phase
Randomly alter bits (the most basic unit of a chromosome) on selected
chromosomes.
6
Evaluate fitness of genetic population
7
Repeat steps 3-6 many times
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Implementation of GA
Defining Fitness Function Terms in Example
Intervention Goals: Prevalence of vitamin A and retinol adequacy of 50% and 85%,
respectively.
f (α1 , α2 , . . . , αK |0.5, 0.85) =
K
X
(ck ∗ αk )
k=1
+ 1000000000(|Estimated prevalence of vitamin A adequacy − 0.5|
+ |Estimated prevalence of retinol adequacy − 0.85|)
Food Vehicle
Sugar (α1 )
Vegetable Oil (α2 )
Wheat Flour (α3 )
Maize Flour (α4 )
Fortification Limit
(µg/g food vehicle)
10.0
25.0
2.4
0.8
ck ($/MT)
0.93
0.26
0.93
0.93
Cost ($/MT)
1.72 + 0.93 ∗ α1
0.26 ∗ α2
8.06 + 0.93 ∗ α3
6.99 + 0.93 ∗ α4
Table: The lower limit for all food vehicles is 0 µg/g. Costs supplied by the Academy of
Educational Development. Notice, all costs are of the form fixed cost plus incremental cost,
where the fixed portion for vegetable oil is $0, and are measured in U.S. dollars per metric ton.
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Implementation of GA
GA Results
Vegetable Oil
Wheat
Cost
2.4
31.0
2.2
30.8
2.0
30.6
11
10
9
8
7
●
Cost ($/MT)
Additional Retinol ( µg/g)
Additional Retinol ( µg/g)
12
1.8
1.6
30.4
30.2
1.4
30.0
1.2
29.8
●
Pop 100 Gen 30 Pop 400 Gen 75
GA Initial Conditions
Pop 100 Gen 30 Pop 400 Gen 75
GA Initial Conditions
Pop 100 Gen 30Pop 400 Gen 75
GA Initial Conditions
Figure: Comparative results from 50 runs of the GA under two different sets of initial
conditions: initial genetic population of 100 was run for 30 generations and an initial genetic
population of 400 was run for 75 generations.
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Prevalence of Nutrient Adequacy vs. Cost Graph
Prevalence of Adequacy vs. Cost Graph
What if no goal was set at the intervention level?
Can run the GA over a grid of intervention goals (δEAR s and δUL s).
Prevalence of Nutrient Adequacy vs. Cost
'Best' Fortification Plans
Prevalence of Nutrient Adequacy vs. Cost
All Fortification Plans
EAR
Prevalence of Nutrient Adequacy (%)
80
60
40
20
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20
25
30
35
UL
1.0
Nutrient
●
Vitamin A
●
Retinol
Prevalence of Nutrient Adequacy (%)
100
0.8
0.6
Nutrient
Vitamin A
Retinol
0.4
0.2
20
25
30
35
20
25
30
35
Cost ($/MT)
Cost ($/MT)
Figure:
Left: Scatter plot of 10,000 “best” fortification plans under all combinations of
δEAR and δUL ∈ {0.00, 0.01, 0.02, . . . , 0.98, 0.99} with initial genetic population size of 50
run for 20 generations. Right: Conditionally “best” fortification plans. As cost increases,
estimated prevalence of vitamin A adequacy increases but the prevalence of retinol
adequacy decreases.
Dave Osthus (ISU)
A GA Optimization Approach
March 14, 2011
24 / 27
Prevalence of Nutrient Adequacy vs. Cost Graph
Variability Considerations
Prevalence of Nutrient Adequacy vs. Cost
95% Confidence Bands
EAR
UL
Prevalence of Nutrient Adequacy (%)
1.0
0.8
Nutrient
Vitamin A
0.6
Retinol
Confidence_Bands
0.4
Actual Data
95% Confidence Bands
0.2
20
25
30
35
20
25
30
35
Cost ($/MT)
Figure:
The “best” fortification plans with 95% central bootstrap confidence bands. 500
bootstrap samples.
Dave Osthus (ISU)
A GA Optimization Approach
March 14, 2011
25 / 27
Future Work
Future Work
Analytical solution to fitness function?
Analysis of run time vs. initial genetic population size and number of
generations run.
Effect of rounding “optimal” candidate solutions to more convenient
numbers.
Dave Osthus (ISU)
A GA Optimization Approach
March 14, 2011
26 / 27
Special Thanks To:
Dr. Alicia Carriquiry
Todd Campbell
Dr. Omar Dary
Dr. Heike Hofmann
Dr. Vivekananda Roy
Dave Osthus (ISU)
A GA Optimization Approach
March 14, 2011
27 / 27