Cargo Loss in Ferrying Operations

Abstract

IN FERRYING operations of valuable items (e.g., aircraft spare parts needed in a theater of operations during a war), the number of items to be carried in each aircraft is frequently under the control of the planner. By using more aircraft he can make the loading per aircraft smaller and hence possibly reduce the probability of large losses at the risk of increasing the probability of small losses. (The small loading also increases the actual ferrying cost.) Because of the value of the items it may be a good investment to buy this insurance. In making his decision as to how many items to load per aircraft, the planner therefore needs to know, as a function of the loading, the probability of losing any given number of items. Sometimes the mean and variance may give sufficient information, without knowledge of the more laboriously calculated probabilities. Usually however, the distributions are not even approximately symmterical nor even unimodal, so that an adequate approximation using only the mean and variance is not possible. In this paper expressions are given for the mean and the variance of the number of items lost, for the probability of losing a given number of items and for the probability generating functions. The moment generating functions may be found from the probability generating functions by substituting ek for 4. For models 1, 3, and 4 several formulas, in addition to the probability generating function and the general formula (m.4) are given for the probabilities of losing exactly a items.' These special formulas are useful for certain values of a. Apparently this problem has not been previously considered in this form in the statistical or operational research literature. We denote by k the number of flights to be used in the ferrying operation. No assumptions are required as to how many aircraft are used in the operations; there may be k aircraft, each making one trip, or fewer aircraft may be used, some or all making more than one trip. Each aircraft carries r valuable items, so that a total of N = kr items are ferried. Each item is assumed to be equipped with a life raft or some other means of preventing it from sinking in case the aircraft