Table 1.
Mathematical symbols used in the model section and their definitions.
Fig 1.
Heterogeneous overordering patterns emerge from strategic interactions among the firms in a supply chain.
Panel (a) presents the evolution of the overordering rates of the 29 firms of the supply chain displayed in panel (b). Firms start without overordering, then adjust their rates to increase their profits. The thick black curve shows the average overordering rate. Each of the other curves corresponds to a firm; they are colored according to the final overordering rate adopted by the firm, as shown by the color bar on the left. In panel (b), firms are nodes, colored to correspond to the curves of panel (a). The 61 grey arrows represent supplier–buyer interactions, while the black arrows indicate the flows that go in and out of the chain: inflows of raw materials and outflows of final goods. The vertical positions of the firms are proportional to their number of total suppliers. Panel (c) uses the same vertical axis to compare the results for this specific supply chain with statistics from an ensemble of 2,000 random supply chains with the same number of firms and connectivity. Crosses indicate means and whiskered bars indicate interquartile intervals.
Fig 2.
Overordering diminishes disruption cascades and mitigates risks.
Both panels refer to the supply chain in Fig 1a and 1b. Panel (a) shows how the distribution of the size of disruption cascades changes between the initial state, shown in blue, in which no firms overorder, and the outcome of strategy evolution, shown in green. The thick vertical lines indicate the means, moving from 5.4 down to 2.1 firms. The size of a disruption cascade is the number of firms affected by supply shortages following an external perturbation; the distribution is obtained by perturbing each firm one by one. Panel (b) displays how mitigation success, which measures the relative reduction in indirect losses, changes during the strategy evolution in Fig 1a.
Fig 3.
Supply-chain fragmentation disincentivizes inventories and reduces risk mitigation.
The figure shows how mitigation success changes with fragmentation for six classes of supply chains, defined by the number of firms, n, and the average number of suppliers per firm, c. Fragmentation is defined as (g−1)/(n−1), where g is the number of groups of integrated firms. Each curve shows the average over 20 random supply chains, and for each of them, 10n group configurations are assessed. The dispersion of the results is shown in S7 Fig in S1 File.
Fig 4.
Durable goods facilitate robust risk mitigation.
The three panels present the mitigation success of integrated supply chains according to productivity z, failure rate p, and for three levels of durability: (a) 0%, (b) 50%, and (c) 100%. The 0% contour is the boundary between productive and unproductive situations. The results are averaged over 10 random supply chains with 30 firms and an average of two suppliers per firm.
Fig 5.
Network indicators help allocate inventories.
Panel (a) shows, for a specific supply chain with n = 30 firms and an average of two suppliers per firm, the mitigation success of six indicators. In each subpanel, the n values of the indicator are centered on 0, normalized to a standard deviation of 1, and then mapped into 262 vectors of overordering rates, with the horizontal axis indicating the average rate and the vertical axis the elasticity, each being stepped in 26 levels. For high elasticities, some rates may fall below 0%, implying that some firms should underorder. Such combinations are removed from further study, as indicated by the upper-left white regions. The star symbols pinpoint the maximum mitigation successes. Panel (b) displays the maximum mitigation successes of the six indicators averaged over 200 random supply chains. The whiskered bars indicate the interquartile ranges. These mitigation successes (middle) are contrasted with the uniform allocation of overordering rates (left), whereby all firms implement the same rate, and with the outcomes of decentralized strategy evolution (right), whereby all firms maximize their own profits.