## Abstract

The optimal expenditure pattern for a double-path engineering project, i.e., a project composed of a nonroutine risky R&D path and a routine nonrisky preparatory path, manufacturing related or marketing related, is studied via the calculus of variations to derive a set of twin second-order nonlinear differential equations whose solution yields the optimal joint expenditure. Assuming independence between the risky and nonrisky paths, a constant return per unit time, a gamma-type unimodal conditional-completion density function for the R&D activity, and the principle of diminishing returns on the effort, we find an interesting interplay between the two paths for the peak position and termination of the expenditures. Counterintuitively, we find that the peak expenditure of the R&D path does not necessarily precede that of the preparatory path, although both path expenditure peaks obey the well-known Kamien-Schwartz theorem. That is, for both paths, the expenditure peak positions precede always the peak of the conditional-completion density function of the R&D path.

Original language | English |
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Article number | 222811 |

Pages (from-to) | 441-455 |

Number of pages | 15 |

Journal | Journal of Optimization Theory and Applications |

Volume | 105 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2000 |

## Keywords

- Calculus of variations
- Expenditure patterns
- Optimal control
- Research and development

## ASJC Scopus subject areas

- Management Science and Operations Research
- Control and Optimization
- Applied Mathematics