David Francas

Capacity investment and the value of operational flexibility in manufacturing systems with product blending

A distinct feature of process industries such as food, chemical and consumer packaged goods is the blending of intermediates into finished goods. In the context of such manufacturing systems the levels of different inputs that can be blended to process a final good define the range of flexibility. Likewise, the cost for using (blending) different inputs defines the mobility element of flexibility. In this paper, we investigate capacity investment and the value of flexibility in the presence of such product blending constraints. We are motivated by recent case studies of food manufacturers, in particular, those manufacturers that seek to increase flexibility via blending of intermediates. We analyse stochastic programs under demand uncertainty of such manufacturing systems. We provide analytical insights into trade-offs when range and mobility are interdependent. Our analytical work gives structural insights into subtle complementarity and substitution effects between dedicated and shared resources in the presence of blending. We analytically show that there is a degradation in the cost performance of such systems with an increase in correlation. We characterise the optimal blending fraction that balances the benefits of higher range with higher costs (lower mobility). Our numerical work shows that a moderate level of blending can significantly improve flexibility and that well-known guidelines for designing limited flexibility change in the presence of blending. For example, blending, even if optimally designed, weakens the appeal of chaining configurations. Overall our work guides resource configuration in industries where product blending is an integral part of the production process.

Lineare Optimierung mit Excel

Lineare Optimierungsprobleme konnen mit Microsoft Excel unter Verwendung von separaten Zusatzprogrammen (sogenannten Add-Ins) gelost werden. Im Lieferumfang von Excel ist bereits das Solver Add-In enthalten, welches in diesem Kapitel zum Modellieren und Losen von linearen Programmen verwendet wird.

Allgemeine lineare Gleichungssysteme

Im Folgenden beschranken wir uns auf \( (m \times 1) \)-Matrizen, also Spaltenvektoren mit \( m \) Komponenten.

Grundlagen der Matrixrechnung

Dabei bezeichnet \( (m \times n) \) [gesprochen: “m Kreuz n”] die Ordnung der Matrix. Matrizen werden gewohnlich mit lateinischen Grosbuchstaben benannt. Unabhangig von ihrer Ordnung besitzt jede Matrix genau eine Hauptdiagonale, welche alle Komponenten \( a_{ij} \) mit \( i = j \) enthalt.