The Optimal Order to Serve in Certain Servicing Problems

A production and transportation system is considered in which the produced items successively are transported to N different selling places. Transportation to the N places may be ordered in N ! different ways and the effects on the system of different orderings among the selling places when servicing them are studied. If the production capacity is K units per unit of time, and if the N places have associated with them times of transport t 1 , …, t N and functions f 1 , …, f N expressing the demand as depending on the time of arrival, it is possible to obtain the total sale on the form Kg N { g N −1 [… g 1 (0) …]} where the functions g n are defined in terms of K , t n and f n . Thus the problem to determine the order to serve that maximizes the total sale is reduced to the problem of finding the optimal permutation of g N { g N −1 [… g 1 (0) …]}. When deriving criterions of optimality for the total sale and related measures of effectiveness the following principle is used: Let E (1, …, N ) be the effectiveness associated with the ordering 1, …, N and let k n , be some characteristic of place n . If k 1 ≦ ⋯ ≦ k N is sufficient (necessary) for E (1, …, N ) ≧ E (1, …, n − 1, n + 1, n , n + 2, … N ) for n = 1, …, N − 1 then this condition is also sufficient (necessary) for E (1, …, N ) ≧ E ( i 1 , …, i N ) for every permutation i 1 , …, i N of 1, …, N .