Optimum forage allocation through chance-constrained programming

Most resource allocation models developed to aid resource planning have been deterministic; that is, the ecosystem and economic variables are assumed to be known with certainty. It is these elements that present problems concerning risk and uncertainty involved in decision making. The objective of this study is to present a mathematical approach for optimizing resource allocation in management situations in which random events occur. This particular technique of decision analysis is chance-constrained programming. The model makes possible the investigation of risk and uncertainty associated with resource management decision making. Range decision-makers must often stock their range before they are sure of the available forage; thus, the amount of available forage is a random element with which managers must contend. The chance-constrained approach to decision making may be used when such random events occur and when it is not possible to plan exactly for future events. Two parameters are used to adjust the mean value of constraints; these are the standard deviation of the constraint value (S_bi) and the probability term which is specified by the manager (K_αi). The mean values are adjusted by the product K_αiS_bi. In the study reported here, a K_αi value of 0.57 gave results which appeared to have usefulness. The results indicate that the penalties a rancher must assume for over-estimating his carrying capacity are greater than the penalties for underestimating the carrying capacity. With the value of the standard deviation used in the example, numbers of livestock and the corresponding net revenues should be about 22% less than those indicated by average forage production. With greater variation in forage production, the reduction would be greater. By using these values, the chance-constrained approach can meaningfully incorporate random variables into a decision model.