A Stochastic production planning problem

TL;DR

This paper analyzes a stochastic multi-good production planning model, solving the HJB equation explicitly to derive optimal production strategies and their properties, supported by numerical experiments.

Contribution

It stylizes and solves a complex stochastic production model with multiple goods, providing explicit solutions and new insights into optimal production rates.

Findings

Optimal production rates are equal across all goods.

Production rates are independent of certain model parameters.

Rates increase with the total number of goods produced.

Abstract

Stochastic production planning problems were studied in several works; the model with one production good was discussed in [3]. The extension to several economic goods is not a trivial issue as one can see from the recent works [4], [5] and [6]. The following qualitative aspects of the problem are analyzed in [5]; the existence of a solution and its characterization through dynamic programming/HJB equation, as well as the verification (i.e., the solution of the HJB equation yields the optimal production of the goods). In this paper, we stylize the model of [4] and [5] in order to provide some quantitative answers to the problem. This is possible especially because we manage to solve the HJB equation in closed form. Among other results, we find that the optimal production rates are the same across all the goods and they also turn to be independent of some model parameters. Moreover we show that production rates are increasing in the aggregate number of goods produced, and they are also uniformly bounded. Numerical experiments show some patterns of the output.