the evolution of the diet model in managing food systems
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The Evolution of the Diet Model in Managing Food SystemsAuthor(s): Lilly M. LancasterReviewed work(s):Source: Interfaces, Vol. 22, No. 5 (Sep. - Oct., 1992), pp. 59-68Published by: INFORMSStable URL: http://www.jstor.org/stable/25061663 .Accessed: 21/11/2011 08:14
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The Evolution of the Diet Model in Managing Food Systems
Lilly M. Lancaster School of Business Administration and Economics
University of South Carolina at Spartanburg
Spartanburg, South Carolina 29303
The electronic revolution in food-systems management began in the '60s with the formulation of the least-cost-meals mathe
matical programming model. This model served as the basis for the CAMP (Computer Assisted Menu Planning) software de
veloped for mainframe computers. Over the next two decades, researchers introduced techniques for incorporating food pref erences into menu planning. Mini- and micro-computer soft
ware emerged. Since the early '60s, managers have used math
ematical programming to enhance cost and nutrition control
and to increase consumer satisfaction.
Preparing
the best possible meals at
the least possible cost is the stated
objective of food-systems management for
most feeding programs. Menu planning is
a key component since the menu deter
mines food, equipment, and personnel re
quirements [Kotshevar 1975]. The menu
planning process deceives both nutrition
experts and the public by appearing to be a
simple procedure. However, the Food and
Nutrition Board's [1980] recommended
daily allowances (RDAs) identify minimum
intake levels for 29 nutrients. Preventive
health care agencies recommend upper limits for intakes of fat, cholesterol, and
sodium. To the MS/OR researcher, plan
ning menus to satisfy these recommenda
tions represents a problem with more than
30 nutritional constraints and thousands of
variables. Furthermore, nutrition cannot be
the only goal. The consumers' food prefer ences must be considered as well as the
Copyright ? 1992, The Institute of Management Sciences
0091-2102/92/2205/0059$01.25 This paper was refereed.
PROGRAMMING?MATHEMATICAL INDUSTRIES?FOOD
INTERFACES 22: 5 September-October 1992 (pp. 59-68)
LANCASTER
impact of the menu upon personnel, tech
nology, and the budget. The increased availability of computers
and of affordable computerized food nu
trient data bases enhances opportunities for developing mathematical models for
menu planning. Models conceived in the
'40s by economists as linear programming
problems have evolved over three decades
into applications software.
The Foundations
The foundation for the application of
MS/OR techniques in planning diets is
Stigler's [1945] "The cost of subsistence."
Finding food mixtures that satisfy nutri
tional requirements is a problem of solving a linear system of equations with more
variables (foods) then equations (nu
trients). When Dantzig [1963] developed the simplex method for solving linear pro
gramming problems, he used Stigler's data
to illustrate the viability and efficacy of the
technique. Since then, the linear program
ming formulation of the classic diet prob lem has continued to evolve. Smith [1963] tried to produce solutions that incorpo rated more acceptable food mixtures by
adding proportionality constraints and up
per bounds to increase food variety. His
model did not contain rules for converting as purchased (AP) foods into edible por
tions (EP) based on recipes. Least-Cost Menu Planning Models
An increasing segment of the population obtains meals through feeding programs at
institutions, such as hospitals, nursing
homes, schools, and prisons and other
food-systems operations, such as restau
rants, mobile meal programs, and congre
gate dining facilities for the elderly. Man
agers and nutrition experts make decisions
on behalf of the consumers in planning menus. The menu displays the items avail
able for selection, course-by-course, meal
by-meal, or day-by-day. Technically this
decision-making process is called menu
scheduling, the allocation of menu items
on a discrete time scale. If the menu lists
more than one item per course, it is selec
tive. Otherwise it is nonselective, and the
menu completely defines the meals. Mate
rial requirements planning is based on the
menu schedule. The menu items are de
fined by recipes, from which one can de
rive the quantities of ingredients needed to
prepare a portion or multiple portions. The first application of mathematical
programming regarded menu scheduling
A final problem relates to the artistic aspects of menu
planning.
as a sequential, multi-stage optimization
problem. Each stage determined the opti mal combination of menu items for a meal.
The data structures of this model also clari
fied the organization of future food-sys tems management software [Armstrong,
Balintfy, and Sinha 1982]. The least-cost
meal model is an extension of the classic
diet model. It is assumed that the compo nents of meals are fixed portions of menu
items. Menu items are mixtures of foods
defined by recipes. While the variable in
the classic diet model is the food, the vari
able in this model is the menu item. Menu
planning is a decision problem of allocat
ing menu items to meals. The meals have a
presumed course structure and the set of
menu items under consideration are as
INTERFACES 22:5 60
THE DIET MODEL
signed to courses.
Menu planning is seldom limited to one
meal but extends to a sequence of meals or
a menu cycle. Variety plays an important role. The model achieves variety by sepa
rating identical and similar menu items
over time by a minimal separation period based on separation ratings collected from
a target population. By introducing separa tion data, planners can restrict the entry of
menu items into the model in the function
of earlier solutions. For example, if mashed
potatoes are in the solution for the first
The solution to the least-cost
diet is the equivalent of the human dog biscuit.
meal of the menu cycle, a separation rating of three prevents mashed potatoes from
appearing in a solution again until the fifth
meal of the menu cycle. The solution to
the model minimizes cost subject to nutri
tion, course structure, and policy con
straints. Multiple choice programming
problems are solved sequentially by (0,1)
integer programming for each time period of the menu schedule. Balintfy [1964] de
veloped a heuristic algorithm with block
pivoting to solve such problems. In early installations, the computer-gen
erated menus provided a 10 to 30 percent cost savings. Balintfy and Nebel [1966] found that the menus were similar in ac
ceptability to those meals planned by tra
ditional methods. Also, the menu met all
the specified nutrient constraints, whereas
menus planned by previously used meth
ods failed to meet nutrient constraints con
sistently. The proven effectiveness of the
mathematical approach and the growing
availability of computers prepared the way
for full-scale software development and
menu control applications [Gelpi et al.
1972]. Following field trials and refine
ments, Balintfy [1969] introduced an ac
ronym for computer assisted menu plan
ning (CAMP) and the CAMP software.
CAMP, as part of the IBM contributed pro
grams library, has been used as a reference
in the development of commercial systems.
Balintfy [1975] describes CAMP in detail.
Since CAMP remains in the public do
main, the number of applications, modifi
cations, and extensions is unknown. Sev
eral applications have been published. One
implementation in a state hospital pro duced 90-day cycle menus [McNabb 1971] and reduced food costs by five percent. CAMP was also used in its original config uration in at least 18 institutions in the
United States and Europe. In Finland,
Tusmi-Nuri and Immonen [1982] planned menus for government-supported schools
that lowered fat content, satisfied other
nutrient requirements, and reduced food
cost by six to nine percent. CAMP was also
extended to generate fiscal reports for food
cost accounting [Fromm, Moore, and
Hoover 1980] and to manage menus on
line [Hoover and Leonard 1982]. In the
hospitality industry, Wrisley [1982, 1983] used the CAMP concept to develop com
puterized food-planning and control sys tems. Wrisley's system is used commer
cially by managers to forecast sales, control
inventory, and compute recipe and menu
costs.
The CAMP system enjoyed considerable
success in the '60s and '70s despite the
problems common to such MS/OR imple
September-October 1992 61
LANCASTER
mentations. Since the computer dominated
the decision process, many decision mak
ers were suspicious of the computer-gener
ated menus. Lacking formal education in
mathematical programming, they found it
difficult to believe that the computer could
generate nutritionally adequate menus and
often recalculated the nutrient content of
the menus by hand. With cost minimiza
tion, managers often suffered budget cuts
An institution offering a selective menu cannot control
the nutrient content or the cost
of a meal chosen by any individual.
and loss of personnel. In addition, because
solutions frequently relied too heavily on
low-cost menu items, menus often lacked
variety. A final problem related to the ar
tistic aspects of menu planning: color, fla
vor, temperature, and texture combinations
for a meal or intrameal compatability. For
example, a monotone colored meal such as
baked fish, mashed potatoes, yellow
squash, and cantaloupe may not be very
appetizing even though nutrition and cost
are controlled. However, no universally
ac
ceptable programmable rules exist for in
trameal compatability. While the menu schedule determines
which item is served in which meal, a
menu plan determines the frequencies with
which items are served during the entire
menu cycle. Mathematical models produce a menu plan in a single stage if the con
straints are defined for the whole cycle.
Such plans are optimal in terms of the cy
cle constraints but have to be scheduled
into a sequence of meals. The CAMP ap
proach, a multistage solution technique, in
spired a single stage formulation of menu
planning problems. This was accomplished
by setting nutrient requirements for the
entire planning horizon and by converting
separation ratings into upper bounds on
serving frequencies. This model produces the least-cost frequencies of menu items
for scheduling. Although this approach has
not been implemented, the fast solution
and shadow price information were useful
for research. It led to the development of a
linear programming food price index
[Balintfy, Neter, and Wasserman 1970]. Preference Maximization Models
Experience with CAMP led to a second
generation of menu-planning models that
quantified consumer food preferences. Food preferences and the desire for variety have long been recognized. Siegel and
Pilgrim [1958] studied the effect of monot
ony on food acceptance. Zellner [1970] studied food acceptance in relationship to
serving frequency. Rolls et al. [1981]
proved that preference for a food item is
penalized when serving frequency is in
creased beyond a quantifiable point be
cause of the "sensory specific satiety" of
repetitive eating. Benson [1960] suggested a functional relationship between food
preference and serving frequency. Balintfy et al. (1974) performed experiments de
signed to establish the nature of the func
tional relations between the measure of
preference for a menu item and the time
history of previous consumption. The pref erence for a menu item was identified as a
concave nondecreasing function of time
since last consumption (Figure 1).
Hyperbolic and exponential functions
INTERFACES 22:5 62
THE DIET MODEL
\f(t)
Figure 1: The preference-time function repre sents the increase in preference for a portion of a particular menu item while considering the time since it was last consumed. If the item is not eaten for a long time, the prefer ence reaches its highest asymptotic value "a."
The adage "absence makes the heart grow
fonder" applies to our favorite foods. The "a"
parameter is potentially different for each item and is estimated by preference rating surveys. The concavity of the preference-time function implies that there is a time interval, t*, between consecutive consumptions for the
same item or similar items where the time
average preference is maximum. This "ideal"
time interval can also be estimated directly from surveys.
are equally useful to represent the time de
pendence of food preferences. The estima
tion of the parameters of the preference time function (Figure 1) utilized the con
cept of separation ratings, as did the
CAMP system. Preference ratings for each
menu item were also collected from a tar
get population and used in the process. For
example, individuals were asked to state
their preferences for menu items using a
numerical scale ranging from minus three
to plus three (least preferred to most pre
ferred). Then for the same list of items, in
dividuals were asked to specify how many
days should pass between repeat servings of those items, or a
separation rating. Us
ing this data, an objective function was
formulated to maximize preference. For
each meal the most-preferred item combi
nations compete for solution with prefer ence computations including both the pref erence and separation ratings. Balintfy and
Paulus [1986] described this approach, but
the application of the preference-time function to menu planning was bypassed in the '70s. Instead, endeavors were made
to extend the planning horizon of the
model from a meal to a longer time period. Sinha [1974] computed the preferences
regarding repetitive serving of items over a
longer planning horizon and as a result de
veloped the preference-frequency function
(Figure 2) and explored nonlinear pro
gramming concepts (Figure 2). Here, pref erence is expressed in the function of serv
ing frequencies during the length of the
planning horizon. The preference-time function and this preference-frequency function show a one-to-one correspon
dence with parameters estimated using al
most identical procedures [Balintfy and
Melachrinoudis 1982]. The major differ
ence is in the data-collection phase. In
stead of separation ratings, individuals are
asked how frequently they would like to
consume menu items during the fixed-time
interval of one month.
For the preference-frequency function, a
quadratic function is the simplest approxi mation. Sinha used a target consumer pop ulation's preference and frequency ratings
September-October 1992 63
LANCASTER
\9M
tan 1a
W &
Figure 2: The preference-frequency function
represents the total preference accumulated
by consuming a portion of a menu item x
times during a fixed time period T, where T is a long time interval such as 30 days. It is assumed that the item is consumed repeti
tively with t = T/x time separations during
time T. For example, if a consumer responds to a survey question that he likes to consume
fresh peaches five times per 30-day month,
there are t = 30/5 or six five-day time separa
tions for peaches during that month. Conse
quently g(x) =
x.f(T/x) and x* = T/t* is the
optimum serving frequency. The tangent of
the slope at x = 0 is equal to the asymptote of
the preference-time function shown in Figure 1. The value of x* is estimated from the
preference-frequency surveys.
to estimate the parameters of a quadratic
preference-frequency function: g(x) = ax
? bx2 where a is the consumer's preference
for a menu item (same as the a assymptote in Figure 1), x is the serving frequency, and
b is the satiety coefficient (a numerical pen
ality for serving an item too frequently). Total preference for all items in a given
menu cycle is the sum of quadratic prefer
ence-frequency functions. The objective is
to maximize a consumer's total preference
subject to nutrition, structural, and budget constraints. The solution provides optimal
serving frequencies for menu items during the menu cycle. In verification experiments
involving school food service, Balintfy,
Rump, and Sinha [1980] used this ap
proach. They conducted a 12-week dou
ble-blind experiment in which preference maximized menus and control menus
planned by school food-service specialists were served in an
alternating sequence.
Results showed that the preference-maxi mized menus increased student participa tion by almost 10 percent. Plate waste de
creased by as much as 24 percent. Food
costs decreased, and the nutritional ade
quacy of the menu cycle was guaranteed. The control menus planned by specialists did not always meet nutrient standards.
The data base and the quadratic pro
gramming model were also used in food
service-policy-evaluation models. In pref
erence-maximizing models, the food
budget is not minimized but set arbitrarily as a constraint. Consequently, the para
metric change of the food budget produces a corresponding change in the preference
maxima (Figure 3). The lowest point of this
preference-efficiency curve coincides with
the least-cost diet (LCD). The solution to
the least-cost diet is the equivalent of the
human dog biscuit. The combination of
menu items may not be desirable for con
sumption, but nutrition and cost are con
trolled. By updating the food-cost data
base at the time of food purchases, the de
cision maker can always
serve the least
cost meals to meet nutrient standards. In
INTERFACES 22:5 64
THE DIET MODEL
300
g 200
c
fe. 100 "5 ~o
0
-100
Figure 3: The preference efficiency curve rep
resents the total preferences for all menu
items in the objective function of a prefer ence-maximized menu-planning model
where budget and nutrient constraints are
imposed. Any point under the curve is an in
efficient menu plan since more
preferred menu items are available for the same budget
and nutrient constraints. The lowest nutri
tionally feasible point on the curve is the "least cost diet" or LCD. The LCD is the
equivalent of a human dog biscuit where the consumer's food preferences are disregarded and the objective is to feed cheaply and eco
nomically. As the budget is increased para
metrically, the preference maxima of the
model increases. The budget corresponding to
the cost-of-decent-subsistence label is the
point on the curve where the marginal utility with respect to calories is zero. In other
words, when the consumer takes the last bite
of food to satisfy his hunger, he or she still finds the food pleasurable. Other points of in terest are the BBD (best buy diet) and the
CAD (cost of affluent diet). The BBD is the diet that gives the highest preference per dol lar. The CAD represents a diet where exces
sive money is spent on high-cost items, such
as caviar and fresh asparagus, even though
preference is not increasing. It implies that
high-cost items do not necessarily buy more
pleasure for consumers.
budget
contrast, the highest point on the curve,
the cost of the affluent diet (CAD), is at
tained when the budget constraint is no
longer binding. There is no rationale for a
budget to exceed this level. For example, caviar and fresh asparagus are high-cost items but not high-preference items for
many populations, such as primary school
children. Spending more money on these
high-cost items will not increase prefer ence.
Arguments can be made to set interme
diate budget levels. One policy considera
tion sets the budget level at the point where an additional budget unit buys the
most increase in preference, the best buy diet (BBD). At this point, a line from the
origin is tangential to the preference effi
ciency curve. This solution represents
serv
ing the meals that offer the consumers the
most pleasure for the money.
Another approach sets the food budget at the "cost of decent subsistence" (CDS)
[Balintfy 1979b]. Quadratic programming solutions produce dual variables represent
ing the marginal utilities of the constraints.
If the calorie constraint is defined as an
equality, the marginal utility of calories can
be either positive or negative depending
upon the budget. This implies that the last
bite of food consumed to meet energy
needs can cause either a pleasant
or an un
pleasant sensation. The cost of decent sub
sistence represents a unique budget level
where the marginal utility of the calories is
zero. This allows the computation of
equivalent preference-maximized serving
frequencies for different food-preference
profiles.
Taj [1984] extended the application of
quadratic programming using econometric
September-October 1992 65
LANCASTER
methods to estimate utility function coeffi
cients. Taj tested estimation of these coeffi
cients with United States Department of
Agriculture (USDA) data. He estimated a
quadratic utility function for the linear de
mand system it generated, determining coefficients using a specialized stepwise
procedure. Balintfy and Taj [1987] devel
oped USDA family food plans using this
approach. Using the same models, Taj
[1990] computed the marginal utility and
marginal cost of nutrient constraints.
Refinements in scheduling optimal fre
quencies resulted in a modeling concept for planning and scheduling. First, entr?e
items were scheduled with maximum sepa
rations; the remaining courses were filled
with items that were least incompatible with the entr?e [Balintfy et al. 1978]. The
state hospital system of New Jersey ap
plied this concept in a system that included
a centrally controlled, locally optimized
computerized food-management informa
tion system. It saved 10 percent of the
food cost per person per day during the
first year of operation while guaranteeing that nutrition standards for the menus
were satisfied [Balintfy 1979a].
Selective Menu Planning All the models discussed so far are lim
ited to planning nonselective menus. Peo
ple usually prefer selective menus, where
they can choose among several items for
each course. However, an institution offer
ing a selective menu, cannot control either
the nutrient content or the cost of a meal
chosen by any individual consumer. The
institution can overcome some loss of con
trol of nutrition and cost if it knows the
probabilities of the items being chosen. It
can then compute the expected cost and
nutrient composition of the meals [Balintfy and Prekopa 1966].
Gue and Ligett [1966] applied this prin
ciple to selective menu planning. In the
CAMP model, they replaced the entity of a
menu item with the entity of a choice
group of items. They obtained choice prob abilities from choices offered in past
menus. Gue, Liggett, and Cain [1968] also
verified that the heuristic algorithm used in
the CAMP software is the most efficient
for this class of problems. The Gue and Ligett approach ignores the
effect of time upon preference and the ef
fect of preference upon choices. Ho [1978] formulated a stochastic approach to selec
tive menu planning, composing the choice
groups using mathematical techniques that
considered probabilistic preferences. He
updated the preference-probability vectors
using a Markov process prior to deriving
sequential solutions with a preference
maximizing multiple choice programming model.
Either approach controls only the ex
pected value of nutrients available for the
average consumer. Proponents of selectiv
ity argue that it is unnecessary to satisfy nutrient constraints on either a
meal-by
meal or day-by-day basis to meet nutri
tional allowances. However, if a meal or a
day's meals fail to meet nutrient require
ments, nutrient excesses or deficiencies
must be carried over and offset in subse
quent meals planned by the computer. One possible solution to this problem is to
offer meals that meet specified nutritional
constraints as choices. Using a computer to
plan, store, and sequence meal selections
controlled for nutrition and cost may be
the next step in applying mathematical
INTERFACES 22:5 66
THE DIET MODEL
programming to food-systems manage ment [Lancaster 1987].
Summary
Food-systems management is a challeng
ing area for computerization and mathe
matical programming. In 30 years of expe
rience, using mathematical programming models to plan menus and make policy de
cisions has proven cost effective. Gener
ally, these models have reduced food costs
by approximately 10 percent. Using math
ematical-programming approaches, menu
planners have provided menus that meet
nutritional standards set by the govern ment and other feeding programs. Using
traditional methods, menu planners have
fallen short of these standards and con
tinue to do so. As models have been con
structed that include consumer food pref
erences, consumer satisfaction has in
creased. Mathematical programming has
given institutions a basis for setting food
budgets. On a larger scale, the USDA has
used a similar approach in setting the costs
for family food plans. These food plans serve as the basis for the food stamp allot
ments given to individuals and families.
The application of mathematical pro
gramming in food-systems management
has fostered the growth of many comput erized managerial techniques. Using com
puterized systems, managers can standard
ize recipes, control ingredients, forecast fu
ture food requirements, schedule meals,
and purchase and control inventory. Be
cause of such systems, managers have
reexamined the philosophies and policies basic to the industry. Mathematical-pro
gramming techniques permit the modern
food-systems manager to develop efficient
and effective management policies.
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