The problem of canceling mutual debts of enterprises (Q1363303)

In modern complex industries every enterprise is connected with many suppliers and consumers: large enterprises can have hundreds and even thousands of them. The enterprise pays its debts from circulating assets, replenishing them by collecting the consumers' debts. If for some reason the consumers delay their payments or the bank payment operations are done too slowly, then the circulating assets will run out, and the enterprise will not be able to pay its debts. If many enterprises find themselves in such a situation, then a crisis of mutual insolvencies will arise. Although the balance of payments of every separate enterprise, i.e., the sum of money owed to it minus the sum of its debts, may be not large (see the figure) that is, the situation is essentially good, no one has money to pay their debts. We suggest an algorithm of mutual cancelling, which is, in a definite sense, optimal and fast enough to be executed on a PC.