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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