Probabilistic formulation of logistic problem with delivery time limitation

Аuthors

Naumov A. V.

e-mail: naumovav@mail.ru

Abstract

In modern conditions, an urgent problem is to modify the classic statements of logistics problem in case of using various means of delivery, for example, small aircraft in conditions of limited delivery time, which are an uncontrolled factor, for example, due to the presence of opposition. The paper considers the logistic task in classical terms of minimizing the cost of delivering cargo from warehouses to destinations, but in conditions of randomness of delivery time. The problem is formulated in terms of stochastic linear programming with a probabilistic constraint on delivery time. This problem into account restrictions on the volume of warehouses and the carrying capacity of delivery vehicles. It is contemplated that the path of the delivery means may comprise multiple delivery points. The travel time from the warehouse to the first delivery point and between delivery points is modeled by independent random variables with a discrete distribution law. Disclosed is an efficient algorithm for calculating probability of simultaneous fulfillment of all restrictions on delivery time to delivery points along a selected trajectory. This algorithm can be used in advance for all trajectories of interest, which reduces the time for making a logistics decision online. This makes it possible to reduce the initial problem of finding the optimal logistic solution by the criterion of minimum costs, taking into account the probabilistic limitation on the delivery time, to the deterministic problem of linear programming with Boolean variables. The results of a numerical experiment illustrating the efficiency of the algorithm for calculating the probability of timely delivery when moving along a selected trajectory are presented.

Keywords:

logistic problem; stochastic linear programming; probabilistic constraint

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