# Multi-Objective Pharmaceutical Portfolio Optimization under Uncertainty of Cost and Return

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## Abstract

**:**

## 1. Introduction

## 2. Goal-Programming with Data Uncertainty

## 3. Pharmaceutical Portfolio Optimization under Uncertainty of Cost and Return

- The total costs are not more than the annual budget at least within the defined confidence level.
- The total return is not less than target return at least within the defined confidence level.

## 4. Numerical Results for Pharmaceutical Project Portfolio Selection

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NPV | Net Present Value |

MICCG | Big-M, Integer, Chance Constrained Goal programming |

## Appendix A

## References

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Scenario 1—input | ||
---|---|---|

Weights | ${w}_{1}^{+}$ (budget) | 1 |

${w}_{2}^{-}$ (return) | 1 | |

Confidence levels | ${\beta}_{j}$ (budget) | $80\%$ (each year) |

$\alpha $ (return) | $75\%$ | |

Return target (NPV) | R | 3500 MUSD |

Allocated budget | ${B}_{j}$ | 150 MUSD per year |

Scenario 1—output | ||

Budget deviation $\left({d}_{1}^{+}\right)$ | $[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$ | |

Budget confidence level (${\beta}_{j}$) | [$100\%$,$98\%$,$83\%$,$93\%$,$98\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$, | |

$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$] | ||

Return target deviation $\left({d}_{2}^{-}\right)$ | 0 | |

Return target | $75\%$ | |

Confidence level ($\alpha $) | ||

Selected projects | ’SUM01-PC’, ’SUM02-Ph3’, ’SUM03-Ph3’, ’SUM04-Ph2’, | |

’SUM06-Ph1’, ’SUM09-Ph1’, ’SUM10-Ph1’, ’SUM11-Ph2’, | ||

’SUM13-Ph2’, ’SUM15-Ph1’, ’SUM16-Ph1’, ’SUM22-Ph2’, | ||

’SUM23-Ph1’, |

Scenario 2—input | ||
---|---|---|

Weights | ${w}_{1}^{+}$ (budget) | 1 |

${w}_{2}^{-}$ (return) | 1 | |

Confidence levels | ${\beta}_{j}$ (budget) | $80\%$ (each year) |

$\alpha $ (return) | $75\%$ | |

Return target (NPV) | R | 4500 MUSD |

Allocated budget | ${B}_{j}$ | 150 MUSD per year |

Scenario 2—output | ||

Budget deviation$\left({d}_{1}^{+}\right)$ | $[0,7.9,42.7,25.7,3.3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$ | |

Budget confidence level (${\beta}_{j}$) | [$99\%$,$80\%$,$80\%$,$80\%$,$80\%$,$86\%$,$95\%$,$100\%$,$100\%$,$100\%$, | |

$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$] | ||

Return target deviation $\left({d}_{2}^{-}\right)$ | 351.6 | |

Return target | $75\%$ | |

Confidence level ($\alpha $) | ||

Selected projects | ’SUM01-PC’, ’SUM02-Ph3’, ’SUM03-Ph3’, ’SUM04-Ph2’, | |

’SUM05-Ph1’, ’SUM06-Ph1’, ’SUM08-PC’, ’SUM09-Ph1’, | ||

’SUM10-Ph1’, ’SUM11-Ph2’, ’SUM14-Ph1’, ’SUM15-Ph1’, | ||

’SUM16-Ph1’, ’SUM22-Ph2’, ’SUM23-Ph1’, ’SUM24-Ph1’, | ||

’SUM25-Ph1’,’Generator 8’ |

Scenario 3—input | ||
---|---|---|

Weights | ${w}_{1}^{+}$ (budget) | 4 |

${w}_{2}^{-}$ (return) | 1 | |

Confidence levels | ${\beta}_{j}$ (budget) | $80\%$ (each year) |

$\alpha $ (return) | $75\%$ | |

Return target (NPV) | R | 4500 MUSD |

Allocated budget | ${B}_{j}$ | 150 MUSD per year |

Scenario 3—output | ||

Budget deviation$\left({d}_{1}^{+}\right)$ | $[0,5.4,38,18.3,0.80,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$ | |

Budget confidence level (${\beta}_{j}$) | [$99\%$,$80\%$,$80\%$,$80\%$,$80\%$,$87\%$,$96\%$,$100\%$,$100\%$,$100\%$, | |

$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$,$100\%$] | ||

Return target deviation $\left({d}_{2}^{-}\right)$ | 415 | |

Return target | $75\%$ | |

Confidence level ($\alpha $) | ||

Selected projects | ’SUM01-PC’, ’SUM02-Ph3’, ’SUM03-Ph3’, ’SUM04-Ph2’, | |

’SUM05-Ph1’, ’SUM06-Ph1’, ’SUM08-PC’, ’SUM09-Ph1’, | ||

’SUM10-Ph1’, ’SUM11-Ph2’, ’SUM14-Ph1’, ’SUM15-Ph1’, | ||

’SUM16-Ph1’, ’SUM22-Ph2’, ’SUM23-Ph1’,’SUM24-Ph1’, | ||

’SUM25-Ph1’ |

${\mathit{w}}_{1}^{+}$ | Budget Deviation | Altered Target Return (NPV) |
---|---|---|

1 | $[0,7.9,42.7,25.7,3.3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$ | 4148.4 |

4 | $[0,5.4,38,18.3,0.80,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$ | 4085 |

9 | $[0,0.1,19.1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$ | 3727.7 |

10 | $[0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$ | 3558.8 |

31 | $[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]$ | 3530 |

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**MDPI and ACS Style**

Farid, M.; Hallman, H.; Palmblad, M.; Vänngård, J.
Multi-Objective Pharmaceutical Portfolio Optimization under Uncertainty of Cost and Return. *Mathematics* **2021**, *9*, 2339.
https://doi.org/10.3390/math9182339

**AMA Style**

Farid M, Hallman H, Palmblad M, Vänngård J.
Multi-Objective Pharmaceutical Portfolio Optimization under Uncertainty of Cost and Return. *Mathematics*. 2021; 9(18):2339.
https://doi.org/10.3390/math9182339

**Chicago/Turabian Style**

Farid, Mahboubeh, Hampus Hallman, Mikael Palmblad, and Johannes Vänngård.
2021. "Multi-Objective Pharmaceutical Portfolio Optimization under Uncertainty of Cost and Return" *Mathematics* 9, no. 18: 2339.
https://doi.org/10.3390/math9182339