An Algorithm for Global Optimal Strategies and Returns in One Fell Swoop, for a Class of Stationary Equipment Replacement Problems with Age Transition Perspectives, Based on Nonzero Starting Ages

Abstract

Aim: This research article aimed at formulating and designing an Excel automated solution-based algorithm for the optimal policy prescriptions and the corresponding returns for all feasible nonzero starting ages in one fell swoop, for a class of equipment replacement problems with stationary pertinent data. Methodology: The aim was achieved by the exploitation of the structure of the states given as functions of the decision periods, and the use of starting age index one, in age-transition dynamic programming recursions, combined with dexterous reasoning regarding the implicit dependence of the dynamic programming recursions on stage numbers. Finally, the article deployed the template to obtain alternate batch optimal replacement strategies for some problem instances, with horizon lengths of 2 to 12 years, and the full set of nonzero starting ages. Results: The investigation revealed that if m is a fixed replacement age in a base problem with horizon length n, and a single starting age t₁ ∈{1,2,...,m } then the optimal solutions and corresponding rewards for the n₂ - stage problem from stage 1+n₂-n to stage n₂ coincide with those of the base problem, not only for the single starting age, but for the entire set of feasible nonzero starting ages in stage 1+n₂-n of the n₂- stage problem. By an appeal to the structure of the states at each stage and the deployment of the preliminary starting age t₁=1 master stroke in the n₂- stage problem, the optimal policy prescriptions and rewards for the base problem for the full set {1,2,...,m } of feasible starting ages coincide with those of the n₂- stage problem from stage 1+n₂-n to stage n₂, resulting in m different problems being solved at once. Conclusion: If n<n₂, such that 1+n₂-n ≥ m, then D*j ( ), ( ) j j j j D s f s are stage j optimal decisions and reward from the template with horizon length 2 2 2 n , for j Î{n +1- n,L, n } if and only if2* ( ) j n n j D s + -2 and ( ) j n n j f s + - are the corresponding optimal decisions and reward in stage 2 j + n - n for the template with the horizon length n .