Programming with Consideration of Variations in Input Coefficients

M. M. Babbar, Gerhard Tintner, and Earl O. Heady’s 1955 journal article develops an exploratory method for attaching probability statements to agricultural production plans obtained by linear programming. Its central distinction is between finding an optimum with average input coefficients and estimating how actual outcomes may depart from that optimum. The article proceeds from an Iowa cropping problem, through statistical predictions for production and returns, to a mathematical appendix explaining the calculations. Linear programming’s capacity to handle complex resource allocations is its starting point; uncertainty about the coefficients is the problem it seeks to address.

If one is interested in knowing the possible range or distribution of possible outcomes, he must consider the possible fluctuations in the input/output coefficients.

The authors distinguish farming from industrial operations whose resource requirements per unit of output are comparatively predictable. Crop yields depend on rainfall, temperature, insects, and other conditions beyond the farmer’s control. Actual resource allocations can also depart from the plan: labor or capital initially assigned to one crop may be diverted to another. These exogenous and endogenous variations interact, changing both resource requirements per bushel and the profitability of alternative plans. The authors therefore treat coefficients as quantities with variation, rather than regarding an optimum calculated from their means as a sufficient guide to production.

The empirical foundation is annual yield data for corn, oats, soybeans, flax, and wheat in Ellsworth Township, Hancock County, Iowa, for 1928–52. Trends are removed to express historical yields on the basis of current techniques, while retaining deviations for estimating variability. A budgeted farming method supplies capital and monthly labor requirements. Labor in each month constitutes a separate resource, although only May, July, and August prove potentially restrictive. Average coefficients determine the initial optimum; annual coefficients provide variance estimates for the subsequent probability calculations. Prices remain fixed at Iowa’s 1952 averages, explicitly leaving a major source of agricultural uncertainty untreated.

The illustrative farm has 148 acres, $1,800 in expense capital, 182 hours of May labor, and 234 hours each in July and August. Applying the simplex method produces an optimum of 3,464.5 bushels of corn and 686.7 bushels of flax, with expected gross returns of $8,021. May and July labor are fully employed, while some land, capital, and August labor remain unused. This result shows that seasonal bottlenecks, rather than the exhaustion of every resource, determine the profitable crop combination. Maximizing gross revenue also maximizes net revenue under the authors’ stipulated treatment of resource supplies as fixed.

Under other yield and input coefficients, however, different programs and profit outcomes are possible.

This qualification marks the transition from optimization to statistical prediction. The authors ask what probabilities attach to different production and profit outcomes when the mean-coefficient program is put into effect. Their procedure assumes that coefficient errors are random, small relative to the coefficients themselves, and normally distributed with zero means and known variances. In the example, three estimated variances suffice: the corn labor coefficient for May and the corn and flax labor coefficients for July. The analysis thus concentrates on variation associated with the resources binding in the mean solution.

The reported 95 percent limits are 2,088–10,282 bushels for corn, 46.5–1,625 bushels for flax, and $6,298–$18,866 for gross profit. Their asymmetry is central to the interpretation: the lower profit limit lies only $1,723 below the mean-program return, whereas the upper limit lies much farther above it. The authors suggest that knowing this downside may increase a farmer’s confidence in adopting the plan. They also calculate probabilities for particular thresholds, including approximately 2.8 percent for gross profit of $6,500 or less and 17 percent for gross profit of at least $10,000. Uncertainty becomes a set of decision-relevant statements rather than an unspecified reservation about the optimum.

The article acknowledges that its procedure does not describe the joint distribution of corn and flax production. It instead examines their combined revenue to assess the simultaneous occurrence of very low outputs. More broadly, its proposed risk comparison is conditional on equal profitability:

If more than one optimum program can yield equal profit, the one that gives the highest lower limit, at a stated level of probability, indicates the program that is “least risky” (or, contrariwise, the program that stands to give the greatest return at a given probability level).

The conceptual move is to supplement the profit optimum with a probabilistic lower bound, not to construct a general trade-off between expected profit and risk. The appendix develops this approach from the equations defining the active and disposal activities. Determinants and cofactors provide expressions for activity levels; variance and covariance calculations support confidence limits and threshold probabilities for outputs and the linear revenue function. The general framework also contemplates variation in resource supplies and prices, although the numerical application holds them constant. Some displayed formulas in the supplied text are incomplete, notably an empty numerator in the final correlation expression, so those expressions require editorial verification.

Again, the reader should be reminded that the above analysis is subject to these assumptions: the errors in the input coefficients are random, small and normally distributed.

This closing reminder bounds the article’s contribution. Its probability statements concern perturbations around the selected program and depend on restrictive distributional assumptions; they do not establish that the same crop combination remains optimal under every realization. Its enduring relevance lies in distinguishing a technically optimal plan from a sufficiently understood planning prospect: historical variability can inform judgments about attainable returns and downside exposure even when prices and resource stocks are treated as fixed.