Research Methods in Nutritional Anthropology (United Nations University, 1989, 201 p.) 
6. Elementary mathematical models and statistical methods for nutritional anthropology 

Decisionmaking models have been employed in a number of anthropological studies to describe the process by which observed distributions of behaviour are generated. From this perspective, individuals in all societies are seen to confront an array of problematic situations that require decisionmaking under conditions of varying degrees of uncertainty and risk. The solution sets they arrive at, in the form of the strategic choices they make, are assumed to be both rational and optimal within bounds. That is, they tend to be the best possible courses of action given available knowledge of alternatives, perceptions of a situation, and evaluations of surrounding constraints. Accordingly, observed behaviour patterns, frequencies, and regularities are conceived to represent the outcomes of a multiplicity of individual choices. The goal of inquiry is to discover and analyse the values and constraints that condition the actual decisions made, and in effect produce the patterns or the set of frequencies of the alternatives observed.
Toward this goal, several methodological paths are open. An especially promising one is to use mathematical models of optimization processes (e.g. linear programming, gametheory, and decision analysis). These supply both a logic and calculus for formulating and analysing peoples' values and goals as optimization criteria. And as White (1974, pp. 401402) observes, they "provide a means of examining the predictions of different axiomatic models of optimizing behaviour, in comparison with behavioural outcomes or statistical distributions of behaviour within a population."
Several important foodrelated activities involve individual and smallgroup (e.g. domesticgroup) decisionmaking concerning such questions as: What foods should be procured? How frequently and in what quantities should certain foods be eaten? Even in relatively restricted situations, questions like these occur and require some choice between alternatives. Institutional diets, such as "dorm food," allow some consumer choice and are at least decided upon by someone (too often not the consumer). Likewise, cultural traditions and norms and seasonal scarcity and poverty can sharply curtail the range of choice, but some options are normally available. In brief, much nutrition behaviour is optative, and in relation to this fact some important questions arise for both the analyst and applied nutrition specialist. These include: What factors determine the specific choices made? What is the nature of the decisionmaking process? Do the foods procured represent maximum nutritional gains and optimal resource allocations (land, labour, wealth)? Is the selected diet the best combination of foods for simultaneously satisfying nutrient need and culturally constituted preferences at minimum cost?
Linear Programming
In this section we introduce linear programming, a mathematical technique for performing optimization analysis, which has been successfully applied to a large and diverse number of nutritional problems both inside anthropology and outside the field. A very useful introduction to the literature in anthropology can be found in Reidhead (1979). Outside of anthropology there are two notable studies clearly pertinent to nutritional anthropology. One, by the geographer Peter Gould (Gould and Sparks, 1969), uses linear programming to construct diets for Guatemala that are minimal in cost, culturally appropriate, and nutritionally adequate. Mapping regional variations in dietary costsurfaces, he demonstrates the potential utility of linear programming for regional planning and family income determination. The second, by the economist Victor Smith (1964, 1975), is an excellent example of the comprehensive use of mathematical programming (linear and nonlinear) for constructing optimal diets and foodproduction systems in Nigeria. His work is also remarkable for an abundance of procedural guidelines.
Broadly speaking, linear programming is a determinate decisionmaking model for finding the most favourable strategy to secure some preferred outcome. It is normally applied when we wish to find nonnegative values of variables that optimize some objective (that is, that maximize the gains and minimize the losses in the process of achieving a goal), when the choice among strategies is restricted by one or more constraints. If the objective can be approximated by a linear function and the constraints expressed as linear equalities or inequalities, then the problem can be resolved mathematically with linear programming. Space prohibits discussing other kinds of mathematical programming (Spivey, 1962; Strum, 1972; Dantzig, 1963). As an illustration, and in the simple examples to follow, we will want to use nonnegative values of two variables x and y that maximize or minimize an objective function of the form
(54) f(x, y) = ax + by
where a and b are given real numbers subject to certain constraints expressible as linear inequalities ( <; > ) in x and y.
Example 5
The first example concerns maximizing an objective function. The data come from an inquiry we are making into the dietary patterns and preferences of middleincome Americans. A 40yearold male wanted to know the best combination of meats to eat in order to get the most protein, with the fewest calories, at the least cost. Beef roast and pork roast were the meats he liked the most. He was willing to spend up to, but no more than, $1.50 for meat for one daily meal. Realizing that he needed about 2,800 calories daily and most of these would come from other ingestibles (e.g. beer), he wanted up to, but no more than, onefourth of his required calories to come from meat (700 car). In July 1980, at the store where he shopped, a heel of round, lean and fat beef roast cost $2.39 per pound (14.9 cents per ounce), and pork roast cost $1.39 per pound (8.9 cents per ounce). A table of nutritive values of foods (Chancy and Ross, 1971, pp. 434469) showed that per ounce, the beef roast contained about 8.33 g protein and 55 cal and the pork roast about 7 g protein and 103 cal.
To formulate his question as a linear programming problem we rephrase it to read: "How many ounces of beef roast and pork roast should he purchase (and eat) to obtain the maximum number of grams of protein?" Since there are two variables, we let x = the number of ounces of beef roast, and y = the number of ounces of pork roast. The objective function to be maximized is given by the linear function z = f (fx. y).
(55) max z = 8.33x + 7y
This is the number of grams of protein in each ounce of beef roast (x) and pork roast (y). Since he wants to spend no more than 51.5(), and beef is 14.9 cents per ounce and pork 8.69 cents per ounce:
(56) 14.9x + 8.69y < 150
Likewise there is a constraint on the total number of calories he wants from these meats, so that
(57) 55x + 103y < 700
which is the number of calories per ounce in beef (x) and pork (y).
Finally, since negative amounts of meat are irrelevant, we write a nonnegativity constraint that
(58) x, y > 0
Since only two variables are involved, a simple and convenient technique for finding the optimal solution to this problem is to use a graphical method. With more than three variables an algebraic method called the simplex algorithm, or some variation of it, is usually necessary (Strum. 1972). In the graphical method we regard an ordered pair (x, y) as rectangular coordinates of a point in the x, y plane. We then graph the equational form of each inequality constraint (56) and (57). We do so only in the positive quadrant (I) because of the nonnegativity requirement (58) on x and y. For example, the equational form of inequality (57) is
(59) 55x + 103y = 7(X)
which can be graphed by rewriting it
(60) y = 6.796  .534x
Then, real numbers can be substituted for x, and the values of y can be plotted. The same is done for (56). This describes the two straight lines which intersect in figure 2. The shaded area satisfies the inequality constraints (5658) and contains an infinite set of solutions within it and on the boundaries. The shaded region has four corner (extreme) points labelled, 0, A, B. and C. Each of these is a basic feasible solution.
An extreme point theorem can be proved to show that the objective function (55) will attain a maximum value, subject to constraints (5658), when the x, y values that define one of these corner points are inserted into it. The insertion of the x, y values defines the optimal solution. At corner point 0, x = 0 and y = 0 (eat neither) and we ignore it because, while it is a feasible constraint, it produces no protein. At corner point A, x = 0 and y = 6.796. Substituting these values for x and y into (55) produces 47.57 g protein. At corner point C, x = 10.067 and y = 0, and substituting these values into (55) produces 83.858 g protein. Finally, at B. x = 8.66 and y = 2.06, and these values, when substituted into (55), produce the maximum value of z, which is 88.2 g protein. (Notice that it is usually necessary to solve the constraint equations simultaneously to define this point exactly.) Thus, we conclude that the optimal strategy for this person is to eat 8.86 ounces of beef roast and 2.06 ounces of port roast per day, which will provide 699.5 calories and cost $1.50. Another way to phrase this solution is to say that about 81 per cent of his meat should be beef roast and 19 per cent pork roast.
Records kept on this person for 43 days reveal that in fact he ate beef and pork for 32 days, and that 53 per cent of his evening meals were beef and 47 per cent pork, although not always in the form of roasts. More extensive studies comparing how well optimal patterns, deduced from a linear programming model, fit actual, observed patterns can provide important information on who is and is not optimizing. Those who show large disparities from the optimum might be studied further to determine whether this is so, and how they differ in other respects from those closer to the optimum. In an applied setting they might be targeted for benevolent intervention, e.g. nutrition and budgetary education and/or resource supplements.
We should mention that a large amount of additional information can also be obtained from a postoptimality analysis of the solution to linear programming problems. For example' the marginal value of each input constraint can be evaluated. This tells us how much the objective function would change as a result of a unit increase in the constant value of each constraint. In this example, cost has a marginal value of .447, and calories a marginal value of .03. Thus, increasing the cost constraint by I cent increases the protein gain .477 g, whereas increasing the caloric constraint by I cal only adds .03 g protein. This relationship can be made clearer by noting that the objective function z is equal to the sum of the products of the marginal and constant values of each constraint' or
(61) z = .447(150) + .03(2000) = 88.05 g protein
We could also perform a sensitivity analysis to determine what effect changes in coefficient values of the objective function and constraint inequations would have on the optimal solution' and what would happen if one or more constraints were added or removed (Strum, 1972).
Example 6
The next example concerns a costminimization problem and is derived from fieldwork among rural and urban Baganda, the largest ethnicity in Uganda. The main dietary staple of the Baganda is the plantain, which is consumed almost daily, constitutes the main bulk of meals' and is used to manufacture a widely imbibed, mildly intoxicating wine produced on nearly every rural farm. In the major city of Kampala, expenditure studies show that the plantain is the biggest single food purchase made by nonproducing urbanities. An edible 100 g portion of plantain (Platt, 1962), contains 128 calories, 31 g of carbohydrates, and 20 mg of ascorbic acid. However, plantain is a poor source of B vitamins, especially of thiamine (B_{1}) for there is only .05 mg of B_{1} per 100 g portion. Thiamine is essential for the utilization of carbohydrate; it is generally recommended that about .5 mg of thiamine be ingested per 1,000 nonfat calories (Latham, 1965; Passmore et al., 1974). Like many tropical agriculturalists, Baganda ingest a large number of carbohydrate calories, not only from plantains, but also from sweet potatoes, cassava, and yams. One possible source of thiamine in the Bagandan diet is the groundnut, which is the most common ingredient of sauces served with plantains (Rutishauser, 1962), and which is an excellent source not only of thiamine (.5 mg/100 g) but also of protein (15 g/100 g) and calories (332 cal/100 g). However, using groundnuts as a thiamine source has a drawback; they are not extensively grown and are very expensive to purchase.
Linear programming may be useful in answering two questions: (a) What is the optimal, lowestcost combination of plantains and groundnuts for obtaining an adequate amount of both calories and thiamine? (b) How do actual use patterns compare with this optimum?
We will answer these questions from the perspective of a lowincome, unskilled male worker in Kampala earning about 140 shillings per month in 1970, who must purchase his food. We have chosen this perspective since have the most useful statistical data on prices and food expenditures for this group of workers (1970 Statistical Abstract, Statistics Division, Ministry of Planning and Development, Uganda).
With respect to question (a), the objective function "c," to be minimized is
(62) min c = 3x + 20.3y
where x = 100 g of plantain at a cost of 3 cents, and y = 100 g of groundnuts at 20.3 cents (100 cents = 1 shilling = 14 cents US) in Kampala in December 1965. In this problem we want to make the value of "c" as small as possible, subject to the following constraints:
(63) 128x + 332y 2 2000
where 128x = the number of calories in 100 g of plantain, and 332y = the number of calories in 100 g of groundnuts. The constraint value 2,000 represents 80 per cent of the recommended 2,500 calories for an average, active male (weight 55 kg) in East Africa (Latham, 1965, p. 234). We assume that at least 500 additional calories will come from other sources. Constraint (64) specifies that at least one mg of thiamine is needed to utilize 2,000 carbohydrate calories in (63). And .05x = the number of mg of thiamine per 100 g of plantain and .5 may = the number of mg of thiamine per 100 g of groundnuts.
(64) .05x + .5y > 1
Constraint (65) requires that we consider only nonnegative amounts of plantains (x) and groundnuts (y).
(65) x, y > 0
Had we wished, we could have specified positive, non  zero values for x and/or y. For example, because of food preferences we might have restricted the solution to x > 8 and y > 1, which would guarantee that at least 8 x 100 g of plantain and 100 g of groundnuts would be contained in the final solution. Food preferences can be important because often optimal diets from a cost and/or nutritional standpoint are unpalatable. Differences in the values of objective functions when food preferences are put in or left out can provide useful measures of "cultural cost" (Gould and Sparks, 1969).
As before, we graph the equational form of (63) and (64) (fig. 3). Again, only the positive quadrant is used because of (65). Notice that this time the shaded, feasible region is on, and to the right of, the intersecting lines. The basic feasible solutions occur at corner points, labelled A, B. and C. Successively inserting the x, y values at these points in the objective function (62) shows the minimum cost function, c, attains its smallest value at point B .543. The optimal solution at this point is x = 14.093 and y = .5907. Thus, to minimize cost, and get at least 2,000 calories and 1 mg of thiamine, the worker should purchase and consume 14.093 x 100= 1409.3 g of plantains and 5907 x 100 = 59.07 g of groundnuts, at a cost of .543. Rephrasing, we could say that, proportionally, about 96 per cent of his intake should be plantain and 4 per cent groundnuts.
Now let us look at the second question, "How do actual use patterns compare with this optimism? Using the nearest month for which urban foodexpenditure statistics are available (February 1964), we find that 13.45 shillings of the monthly food expediture of a typical unskilled worker in Kampala were spent on plantains, and 3.6 shillings were spent on groundnuts. At the December 1965 prices quoted above, these sums would buy 44.83 kg of plantains and 1.77 kg of groundnuts. The relative proportion of actual use indicated by these purchases is thus 96.2 per cent plantains and 3.8 per cent groundnuts, which is strikingly close to the optimum deduced from the model.
Unfortunately, the only other data we have on actual consumption patterns are from the nutritionist Rutishauser's study (1962) of the composition of meals and recipes in Buganda. In this study it was found that, typically, the ratio is 2 ounces of groundnuts to 28 ounces of plantain per serving. Thus, the relative proportion is 93.3 per cent plantain and 6.7 per cent groundnuts, which is also close to the optimum. If these figures are at all representative, and admittedly they leave a lot to be desired, then it appears that lowincome urban workers in Kampala are probably securing a nutritious mix of calories and thiamine from these sources at near optimum cost.