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 t₁, …, t_N and functions f₁, …, 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[… g₁(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[… g₁(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 k₁ ≦ ⋯ ≦ 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(i₁, …, i_N) for every permutation i₁, …, i_N of 1, …, N.