Local Food Price Analysis by Linear Programming : A New Approach to Assess the Economic Value of Fortified Food Supplements

Local food price analysis by linear programming:
A new approach to assess the economic value of
forti? ed food supplements
André Briend, Elaine Ferguson, and Nicole Darmon
Abstract
Linear programming analysis is a powerful approach
for identifying a low-cost nutritionally adequate diet
Linear programming can be applied to identify a nutri-
[1], but it has been only infrequently used for this
tionally adequate diet of the lowest cost, since price and
purpose in human nutrition [2–4]. This paper also
nutrient contents are linearly related to food weight. Most
illustrates how this technique can be used to estimate
computer spreadsheets now include an easy-to-use solver
the economic benefits expected from the introduction
function that is suitable for this purpose. This approach
of different fortified foods using local food prices. As
can also be used to estimate the effect of introducing a an example, it will compare the economic value of a
food supplement on the minimal cost required to provide
classical blended food with that of a nutrient-dense
a nutritionally adequate diet. It can also provide an spread known as a “foodlet,” a highly fortified food
estimate of the expenses saved by families in relation that can be regarded as a big tablet (foodlet for food +
to the sums spent by the donor after the distribution tablet) [5], for childhood diets in Chad.
of a food supplement. This method is illustrated by
comparing the economic value of two food supplements,
a traditional blended flour and a nutrient-dense spread
Estimation of the minimum price of a
(a “foodlet”) in rural Chad. The limitations of this balanced ration by linear programming
approach and the need to interpret its findings carefully
in relation to field observations are discussed.
Linear programming is a technique that minimizes a
linear function of a set of variables while respecting
multiple linear constraints on these variables. It can
Introduction
therefore be used to minimize the price of a diet while
fulfilling constraints introduced to ensure a palatable,
Food fortification is often advocated on the grounds nutritionally adequate diet. The linear function to be
that it is an inexpensive strategy to increase the nutri-
minimized in this case is the cost of a balanced diet, as
tional value of the diet of the poorest populations. Even
expressed mathematically below:
if the cost of food fortification is low, its implementa-

Cost of ration = W C + W C …+ W C
tion requires a strong commitment from local govern-
1 1
2 2
n n
ments, food industries, and donor agencies, who do where C , C , C are the costs per unit weight of foods
1
2
n
not always perceive the benefits of this approach. In 1 to n.
part, this is because the benefits of food fortification The various constraints needed to ensure a palatable,
are not easily quantified in economic terms.
nutritionally adequate diet include nutritional con-
straints on the minimum energy and nutrient content
André Briend is affiliated with the Institut de Recherche
in the diet, as well as food-consumption constraints on
pour le Développement and is currently working at the
the maximum acceptable daily portions of individual
Institut des Sciences et Techniques de la Nutrition et de
foods. These constraints should be expressed as linear
l’Alimentation in Paris, France. Elaine Ferguson is affiliated
constraints to facilitate the analysis, and are briefly
with the Department of Human Nutrition, University of
Otago, in Dunedin, New Zealand. Nicole Darmon is affiliated
described below.
with the Institut National de la Santé et de la Recherche
An adequate diet can be defined as a diet fulfilling all
Médicale and is currently working at the Institut des Sciences
current nutritional recommendations, with each nutri-
et Techniques de la Nutrition et de l’Alimentation in Paris.
ent being present in amounts equal to or superior to its
Mention of the names of firms and commercial products
recommended daily allowance (RDA). In mathematical
does not imply endorsement by the United Nations University.
184
Food and Nutrition Bulletin, vol. 22, no. 2 © 2001, The United Nations University.
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Local food price analysis by linear programming
185
terms, this can be expressed by a set of inequalities:
options assuming that all constraints are linear and
that all variables are positive should be activated before

N W + N W …+ N W > RDA N
11
1
12
2
1n
n
1
starting the analysis.

………
The solver function of recent spreadsheet versions

N W + N W … + N W > RDA N
j1
1
j2
2
jn
n
j
also accepts nonlinear constraints in the model. Hence,
constraints based on complex mathematical functions
where
can be introduced, giving, for example, a better approx-
n is the number of foods habitually imation of the mineral availability than the phytate/
consumed by the local population;
zinc ratio. Nonlinear programming, however, is a
j is the number of nutrients for which
more complex technique and may in some cases give
an RDA is specified;
nonoptimal solutions, unless the analysis is repeated
W , W …W are the weights of foods 1 to n; and
several times using different initial values for food vari-
1
2
n
N , N , …N are the contents of nutrients 1 to j
ables. Therefore, only linear programming is recom-
11
12
jn per unit of weight of food 1 to n.
mended, at least in the initial stages of an analysis.
All these constraints are linear constraints, because
they consist of a simple sum of products. If the RDA An example based on a food price survey
selected for iron and zinc assumes moderate bioavail-
in Chad
ability, then an additional constraint is required on
the phytate:zinc molar ratio (PZ), whereby the PZ is To illustrate the merits of linear programming for
below 15. This constraint is not linear, because it is identifying low-cost diets, the minimum price of a
based on a ratio. However, it can be expressed as a nutritionally adequate ration for a child aged three to
linear constraint by transforming it to
six years was estimated from a price survey performed
in Mao, Chad. Mao is the main city of Kanem Province
15
Z – P > 0
in a semi-desert part of northern Chad. Only a limited
where Z and P are the molar zinc and phytate contents
number of foods traditionally given to children in this
of the diet.
area were included in the analysis.
This equation can also be transformed into a sum For this analysis, local food prices were converted
of products in the following form:
into prices in US$ per kilogram of the edible portion.
The nutritional constraints used in the analysis were
(15Z – P )•W + (15Z – P )•W
1
1
1
2
2
2
based on the preliminary World Health Organization

… + (15Z – P )•W > 0
n
n
n
(WHO) nutritional recommendations for children of
this age group [6] for all nutrients except calcium. For
where Z and P represent the molar zinc and phytate
calcium, the UK recommendation [7](450 mg/day)
n
n
contents of food n.
was used instead, because an optimized diet was not
If used with only nutritional constraints, linear
realizable with the use of the preliminary WHO rec-
programming can lead to unrealistic diets, since this ommendations for calcium (600 mg/day). An addi-
approach will select a few low-cost, highly nutritious tional constraint was also introduced on the PZ (i.e.,
foods, such as beans, green leafy vegetables, dry fish, < 15) to warrant the assumption of moderate zinc and
or liver, which will not be eaten in large quantities iron availability [8]. These constraints are presented
in practice. To avoid this problem, constraints on in table 1.
maximum daily portion sizes for each food should To ensure that optimized diets were realistic, food
be introduced. Ideally these portion sizes should be constraints were also introduced that limited the
defined by using food-consumption survey data col-
daily portions of available foods (see table 2). Food-
lected from the population of interest or, if this is not
consumption data were not available for three- to
possible, from a similar community.
six-year-old children in Chad, and therefore the 75th
Linear programming is based on a mathematical centiles of observed intakes of three- to six-year-old
iterative approach involving multiple calculations of rural Malawian children were used instead, assuming
products and sums, which can be quickly performed that limits of intake in these two poor rural communi-
by a personal computer. Calculations presented below
ties would be similar. These data were collected by
were done with the Excel 97 (Microsoft) spreadsheet,
using three-day weighed food records in the posthar-
which has a linear programming function called a vest season in rural Malawi [9]. Some adjustments
“solver function” in all its recent versions. This function
were also made to take into account local food avail-
is found in the “tools” menu. For some versions of this
ability in Mao. For example, the 75th centile for fresh
program, this function is not automatically installed. fish intake in Malawi was used to define the food
However, it can be imported from the original CD constraint for fresh meat intake in Chad. Spirulina, a
of the program. It is also important to note that the green-blue alga growing spontaneously in the oasis in
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186
A. Briend et al.
this part of Chad and traditionally collected and dried
were taken from the World food dietary assessment
to prepare a nutrient-rich sauce, was also added to the
system (version 2.0) [11].
food database. During the optimization process, the The diet obtained by cost minimization with local
nutritional contents of all foods except Spirulina [10]
foods is shown in first column of table 2. Most foods
present in the food database were selected, and several,
such as sweet potatoes, eggs, and dried fish, were set at
TABLE 1. Nutritional constraints introduced in the models
the maximum allowed. By definition, this diet fulfils all
for estimating the minimal price of a balanced ration for
three- to six-year-old children in Mao, Chad
nutritional constraints, which means that its nutrient
content is equal to or above the RDA for all nutrients.
Nutrient Amount
This diet should not be regarded as an ideal diet, but
as the lowest-cost nutritionally adequate diet for this
Macronutrients [6]
Energy
> 1550 kcal
population where dietary quality is limited by cost.
Proteins
> 20 g
The most difficult nutritional constraints to fulfill
in this setting were energy, niacin, folate, and PZ, which
Minerals
were at the minimum and maximum levels for the con-
Calcium
> 450 mg [7]
Magnesium
> 76 mg [6]
straints imposed (table 3). This optimized diet should
Zinc
> 4.8 mg [6]
be regarded as a first estimate, and it must be followed
Iron
> 5 mg [6]
by field observations to confirm that the optimized diet
Phytate/zinc
< 15 mg [(8]
is compatible with general food-consumption patterns
Vitamins [6]
of children in the region. Additional constraints can
Vitamin B1
> 0.6 mg
also be added if necessary: for example, constraints on
Vitamin B2
> 0.6 mg
the percentage of energy provided by different food
Vitamin B6
> 0.6 mg
groups to delimitate local food-consumption patterns.
Niacin
> 8 mg
Conversely, some constraints may be relaxed, such as
Folate
> 200 µg
the upper limit on daily portions for highly acceptable
Vitamin B12
> 1.2 µg
foods. As a rule, tightening constraints will increase
Vitamin C
> 30 mg
the estimated minimum price of a balanced ration,
Vitamin A
> 450 µg RE

whereas relaxing them will lower this price.
TABLE 2. Food consumption constraints imposed during optimization, and daily portions of foods selected
for an optimized (low-cost, nutritionally adequate) diet for a three- to six-year-old child, with and without
introduction of traditional blended flour and highly nutrient-dense spread in the linear programming model

Maximum amount

Amount of food selected (g)a
of
food
allowed

during diet
Only local
Food
optimization (g)
foods
Flour addedb
Spread addedc
Spirulina alga
5
5
0 0
Groundnuts
45 0 0 0
Banana 75
75
0 0
Maize flour
255
0
53
255
Millet flour
175
71
23
0
Dried
okra
65 33 0 0
Oil 17 17 17 17
Cowpeas
60 16 6 3
Eggs 40
40
0 0
Onions
4
4
0 0?
Sweet potatoes
165
165 156
6
Dried fish
25
25 25 12
Rice 110
110 110 39
Mutton 80
80 0
36
Nutrient-dense spread
40
NA
NA
26
Blended flour
255
NA
120
NA
Cost
(US$)
0.704 0.329 0.286
a. Foods set at their maximum level by linear programming are in boldface type. Amounts are given for foods as bought from
the market, before preparation. NA, not allowed.
b. Estimated cost: US$1/kg.
c. Estimated cost: US$3/kg.
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Local food price analysis by linear programming
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This example has been presented to demonstrate and distributed in food-supplementation programs in
how linear programming can be used to delineate Chad in recent years.
the minimal cost required to provide a nutritionally To estimate the maximum amount of blended flour
adequate diet for a population of three- to six-year-old
a child of this age can reasonably eat per day, the
children in this part of rural Chad. Such information
75th centile of intake for maize flour in Malawi was
is useful for nutrition education purposes and for chosen [9]. For nutrient-dense spread, it was limited
evaluating economic factors in relation to dietary to 40 g, which was the average intake observed in the
adequacy.
supplementation study in Algeria on children aged
three to five years.
Comparison of the economic impact of two possible
In this example, it was assumed that blended flour
food aid programs in Chad
and nutrient-dense spread could be produced in
N’Djamena, the capital city of Chad, and transported
In some communities, an affordable, nutritionally to Mao at a cost of US$1 and US$3 per kilogram,
adequate diet based on local foods may be difficult respectively. These figures—although realistic—are
to achieve without the introduction of a low-cost used only to illustrate the use of linear programming
fortified-food supplement. In these circumstances, and should not be taken literally. The production costs
linear programming analysis can be used to evaluate for nutrient-dense spread were estimated from the
the effect of alternative food supplements on the price of locally available ingredients and the cost of
price of a ration. To illustrate this application of adding a mineral and vitamin mix.
linear programming, we have compared the economic
Table 2 shows the optimized ration and minimal
value of two food supplements, a traditional blended costs obtained when blended flour and nutrient-dense
flour and a highly nutrient-dense spread, defined as spread were added to the list of foods available for
a “foodlet” [12]. Their nutritional values are shown selection in the ration, using the same constraints as
in table 4. For nutrient-dense spread, the fortification
those previously described (see tables 1 and 2). As
levels chosen were those previously field-tested in outlined above, these rations must be interpreted care-
Algeria [13]. High fortification levels are made possible
fully and ideally should be field-tested for their accept-
in this spread by the attractive taste of peanut, which ability and, if necessary, reanalyzed using additional
can easily hide high levels of unpalatable vitamins constraints.
and minerals. The flour was a blend of maize and To illustrate the use of linear programming for
cowpea flours with sugar, fortified with a standard defining the economic impact of fortified supplements,
mineral and vitamin mix [14]. This formula of this we will assume that the proposed diets (table 2) were
blend is similar to that of the blended flour produced
acceptable and palatable. This analysis showed that
the inclusion of either blended flour or nutrient-dense
spread in the model decreased the estimated minimum
TABLE 3. Nutritional content of diets for a three- to six-year-
old child selected by linear programming analysisa
TABLE 4. Nutritional composition of traditional blended
Only
local
Flour
Spread
flour and highly nutrient-dense spread (per 100 g)
Nutrient foods
added
added
Macronutrients
Nutrient Flour
Spread
Energy (kcal)
1,550 1,550 1,550
Energy (kcal)
330
630
Protein (g)
60
49
40
Protein (g)
12
10
Minerals (mg)
Calcium 546
617
450
Minerals (mg)
Magnesium 179
215
388
Calcium 126
1000
Zinc 7
11
20
Magnesium 58
156
Iron 13
20
16
Iron 12
42
Phytate/zinc < 15 mg 15
15 14
Zinc 6
41
Phytates 544
289
Vitamins
Vitamin B1 (mg)
0.63
0.92
1.88
Vitamins
Vitamin B2 (mg)
0.81
1.01
1.53
Vitamin A (µg RE)
400
2000
Vitamin B6 (mg)
1.3
1.2
1.7
Vitamin C (mg)
30
125
Niacin (mg)
8
14
23
Vitamin B1 (mg)
0.4
3.5
Folate (µg)
200
293
200
Vitamin B2 (mg)
0.5
4
Vitamin B12 (µg)
1.3
1.2
1.2
Vitamin B6 (mg)
0.2
3.5
Vitamin C (mg)
39
64
30
Vitamin B12 (µg)
1
3.5
Vitamin A (µg RE)
1,850
2,130
526
Folic acid (µg)
200
500
Niacin (mg)
6.8
50
a. Nutrients set at their allowed limit by the program are in bold type.

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188
A. Briend et al.
price of a balanced ration (table 2), using quantities principles and selected applications of the approach
less than the maximum limit specified in the program.
have been presented here. It is noteworthy that this
Assuming that these foods are distributed freely (food
method could be further refined by taking into account
aid), the resultant savings for a nutritionally adequate
costs not included in this example, such as the cost
diet per child per family will be equal to the difference
of targeting food distribution, of administrative over-
between the total cost of the diet without and with heads, or of training food aid staff. It could also take
the supplement (both costs shown in table 2) plus the
into account foods grown by the family by giving them
price of the supplement itself. In this example, this an economic value.
saving (D in table 5) is slightly higher for blended flour
The sensitivity of linear programming to selected
than for nutrient-dense spread. In contrast, the ratio constraints, however, is its major weakness. Clearly,
between the amount saved by the families and the this approach should not be used in isolation, and
amount spent by the donors (R in table 5) is higher for
the validity of its conclusions should be field-tested.
nutrient-dense spread than blended flour, because of The chosen set of nutritional constraints should be
the higher cost of blended flour (B in table 5).
based on internationally accepted nutritional recom-
The latter gives a better comparative estimate of the
mendations, such as those published by international
economic value of the alternative food supplements organizations. Ideally, the food-consumption con-
evaluated (i.e., each US$1 spent by the donor on straints used in the model should be derived from
nutrient-dense spread saved US$7.07 for the families,
food-consumption data collected in the community
as compared with US$4.15 for blended flour). In other
of interest. In their absence, constraints based upon
words, in this example, the foodlet would be more survey data collected in a comparable environment,
cost-effective than the traditional blended food. These
as presented in this example, may be a useful starting
results were not easily predictable when the costs were
point for the analysis. Building up an international
compared in isolation (i.e., the prices per kilogram database of food-consumption constraints for different
and per quantity of energy), a result emphasizing the age groups, especially for nutrient-dense foods, would
merits of this type of analysis.
facilitate the application of this method. The validity
This analysis will also show when a proposed program
of these constraints could then be confirmed and if
has a ratio of amount saved to amount spent less necessary adjusted on the basis of a series of simple
than 1, which means that the money saved by the observations in the community itself.
families will be below the amount spent by the donor.
This method has wide applications for different
This is likely to be the case for unfortified blended types of nutrition-intervention programs, including
flour prepared from locally available foods: these foods
supplementation, fortification, and agriculture pro-
are more expensive than the sum of the basic ingredi-
grams. Despite its limitations, it clearly provides useful
ents used in their composition and have no superior information for evaluating the economic benefits of
nutritional value compared with the meal a mother different intervention programs for the poor.
would prepare at home with the same ingredients.
Limitations of the presented approach and possible
Acknowledgments
future applications
The authors wish to thank Mr Radandi Lacsala,
Linear programming is a very powerful tool for analyz-
Centre de Nutrition et de Technologie Alimentaire,
ing the cost of a nutritionally adequate ration prepared
N’Djamena, République du Tchad, for his help during
from different locally available foods. Only the general
the price survey in Mao (Kanem).
TABLE 5. Estimation of the economic impact and efficacy of blended flour and highly nutrient-dense spread

Minimum cost of
Cost of
Estimated cost for
Estimated saving
Ratio of

ration for child
supplement (B)
family of ration
for family to feed
amount saved

with fortified
(price per
for child when
child balanced diet
by family to

supplement (A)
kilogram
food supplement
when supplement
amount spent

(from table 2)
amount in
is given
is given
by donor
Food
(US$)
table 2)(US$)
C = (A–B)(US$)
D = (MEPa – C)(US$)
(R = D/B)
Flour
0.33 0.12 0.21 0.50 4.2
Spread
0.29 0.07 0.22 0.49 7.1
a. MEP, Minimum estimated price for a nutritionally adequate diet for a three- to six-year-old child from local foods: US$0.70.
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