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A theory of supply chains. (English) Zbl 1030.90003
Lecture Notes in Economics and Mathematical Systems. 526. Berlin: Springer. viii, 123 p EUR 32.95/net; \$ 47.00; £23.00; sFr 56.50 (2003).

The author investigates supply chains, i.e., networks where suppliers produce goods for generic customers, possibly visiting intermediate suppliers and being altered or combined in the process. The output of each supplier is determined by a function of its inputs. The size of flows of different goods through the network depends on the customer demand, the flow of orders across suppliers, and on the policies that the suppliers use to place orders and replenish their inventories.

The investigations are mainly restricted to a serial supply system where supplier $j$ receives all its input from supplier $j+1$ and ships all his goods to supplier $j-1$. It is assumed that the order policy of supplier $j$ is based exclusively on the order history of supplier $j-1$ on the one hand and its own status on the other hand. The book investigates the instability of such policies, i.e., the increased variability of order sizes and inventory levels with the supplier as $j$ increases.

For analyzing the stability of order policies the concept of stability in the small and stability in the large (as considered in control theory) are introduced, where the former means that any infinitesimal perturbation in the customer demand input when the system is in a steady state generates only infinitesimal order size and inventory perturbations for any supplier $j$, and the latter means that the order size and the inventory must be bounded for all $j$ at any time.

Necessary and sufficient conditions for policies assuring stability in the small are developed. Moreover, the class of strongly stable policies (being stable in the small and simultaneously in the large) is introduced and algorithms are proposed to determine such a policy in special cases. The techniques are partly taken from the field of traffic flow theory and its its connections to queuing theory are briefly outlined. The theoretic findings are illustrated by means of various examples. Finally, it is shown how to estimate the total cost and arrive at cost-optimal policies. In the concluding chapter some hints for generalizations are given.

##### MSC:
 90B05 Inventory, storage, reservoirs 90-02 Research monographs (optimization) 90B20 Traffic problems