# I. Introduction
Recent global supply-chain problems have received acute attention from every corner on the planet. While ad hoc ex-post analyses are perhaps timely [1], [2], preventative ex-ante treatments are more fundamental. Standard topics of supply chain management include: the bullwhip effect, vertical integration, and point-by-point statistical control [3];this paper seeks to add to the list the critical paths method(CPM)[4], [5] (for applications of CPM, cf. [6], [7]), which bears the common construct of a partially ordered set. In addition, we extend the scope of a directed tree of uni-directional edges/paths connecting all the vertices/nodes [8], [9], to a network of mutually dependent economic agents, which then readily leads to Leontief's inputoutput table for the gross domestic product (GDP) [10].As such, Section 2 below will connect CPM to a supply-chain problem, and Section 3 will show how an input-output analysis can address a global disruption over an economy, where we will incorporate the apparatus of elasticities of substitution, proportional changes in the ratio of two factors due to a change in the ratio of their prices. Section 4 will draw a summary.
# II. Supply Chain by CPM
Let { } { } A 1, 2, , N 1 i a i n ? = ? ? ?
be a set of activities with strict partial order relations ? , where " " j k a a ? denotes j a being a predecessor of k a , and denote the set of all the largest elements of A by L , which have no successors. By Hausdorff maximum principle [11], there exists a maximal simply ordered subset B m of A that has its largest element L m a ? ; i.e., B m is a critical path.
Next, collect all these critical paths { } m B and conduct the usual CPM analysis for each B m [6]; then one arrives at a complete set of optimal solutions for a supplychain problem. .
i j ij k i j ij k k s ? ? < < ? ?? ?
Finally, measure the economic significance of k by
, k k k v ? ? ?
and establish a decreasing sequence
# Notes
As has been observed in recent years, there are mainly three kinds of situations that disrupt a general economy: that due to labor shortage for certain specific tasks, that due to the lack of special components in a production process, and that due to disparate outlet distributions when the same commodity does not have a universal availability; all these problems can be alleviated or even prevented by an ex-ante detailed CPM analysis. The Covid-19 pandemic has inflicted losses of GDP's across countries of different degrees; here we contend that by conducting an input-output analysis as outlined above, the public sectors of an economy may act to balance supply and demand strategically over the markets. In summary, this paper has presented two methodologies for dealing with a supply-chain problem. Future studies might pursue Gantt Charts for all the commonly experienced supply chain problems as well as an estimation of the elasticities of substitution in the frame of the above conducted Leontief input-output analyses.
![of the trading values of economic agent i sold to agent j in GDP. Obtain the row sums which represents the total trade value of k, analogous to the sum of exports and imports of an economy.Fix k as a producer; measure its inputs substitutability by the product of the elements of the upper triangular sub-matrix of ; two measures of substitutability for k as a producer and as a consumer by ( ); ;](image-2.png "?")
![by which one can address an economy-wide supply chain problem more à propos, in particular, paying special attention to the case of .](image-3.png "?")
. Hanaka, T., Kagawa, S., Ono, H., and Kanemoto, K., Finding environmentally critical transmission sectors, transactions, and paths in global supply chain networks, Ener. Econ., 68 (1), 2017, 44-52.
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