Optimization Concepts: II—A More Advanced Level

Abstract

Optimization subjects are addressed from both an introductory and an advanced standpoint. Theoretical subjects and practical issues are focused, conjugating Optimization basics with the implementation of useful tools and supply chain (SC) models. In the prior Introductory chapter on Optimization concepts, Linear Programming, Integer Programming, related models, and other basic notions were treated. Here, the More Advanced chapter is directed to Robust Optimization, complex scheduling and planning applications, thus the reading of the prior Introductory chapter is recommended. Through a generalization approach, scheduling and planning models are enlarged from deterministic to stochastic frameworks and robustness is promoted: model robustness, by reducing the statistical measures of solutions variability; and solutions robustness, by reducing the capacity slackness, the non-used capacity of chemical processes that would imply larger investment costs.

Appendix 1: Technical and Economic Estimators

The non-robust NPV expected value:

$$ Ecsi = \sum\limits_{r = 1}^{NR} {prob_{r} \xi_{r} } $$

(A.1)

The negative deviation expected value:

$$ Edvt = \sum\limits_{r = 1}^{NR} {prob_{r} .dvtn_{r} } $$

(A.2)

The non-satisfied demand expected value:

$$ Ensd = \sum\limits_{r = 1}^{NR} {\frac{{prob_{r} }}{NC.NT}\left( {\sum\limits_{j = 1}^{NC} {\sum\limits_{t = 1}^{NT} {Qns_{jtr} } } } \right)} $$

(A.3)

The capacity slack expected value:

$$ Eslk = \sum\limits_{r = 1}^{NR} {\frac{{prob_{r} }}{M.NC.NT}\sum\limits_{j = 1}^{NC} {\sum\limits_{t = 1}^{NT} {\left\{ {\sum\limits_{i = 1}^{M} {\sum\limits_{s = 1}^{NS} {\sum\limits_{p = 1}^{NP} {p(i).y_{isp} .\left( {dv_{js} - S_{ij} .\frac{{W_{jtr} }}{{n_{jtr} }}} \right)} } } } \right\}} } } $$

(A.4)

The percentage non-satisfied demand expected value:

$$ \% Ensd = \frac{Ensd}{Qmed}.100,\quad {\text{with}}\quad Qmed = \sum\limits_{r = 1}^{NR} {\frac{{prob_{r} }}{NC.NT}\left( {\sum\limits_{j = 1}^{NC} {\sum\limits_{t = 1}^{NT} {Q_{jtr} } } } \right)} $$

(A.5)

The percentage capacity slack expected value:

$$ \% Eslk = \frac{Eslk}{Vtotal}.100,\quad {\text{with}}\quad Vtotal = \sum\limits_{i = 1}^{M} {\sum\limits_{s = 1}^{NS} {\sum\limits_{p = 1}^{NP} {\left( {y_{isp} .dv_{is} } \right)} } } $$

(A.6)