# One difficulty with solving the general minimum-cost flow problem with upper and lower bounds on…

One difficulty with solving the general minimum-cost flow problem with upper and lower bounds on the variables lies in determining an initial basic feasible solution. Show that an initial basic feasible solution to this problem can be determined by solving an appropriate maximum-flow problem. (Hints. (1) Make a variable substitution to eliminate the nonzero lower bounds on the variables. (2) Form a ‘‘super source,’’ connected to all the source nodes, and a ‘‘super sink,’’ connected to all the sink nodes, and maximize the flow from super sink to super source.)

Q153.

As the leader of an oil-exploration drilling venture, you must determine the least-cost selection of 5 out of 10 possible sites. Label the sites S1, S2, . . . , S10, and the exploration costs associated with each as C1,C2, . . . ,C10.

Regional development restrictions are such that:

i) Evaluating sites S1 and S7 will prevent you from exploring site S8.

ii) Evaluating site S3 or S4 prevents you from assessing site S5.

iii) Of the group S5, S6, S7, S8, only two sites may be assessed. Formulate an integer program to determine the minimum-cost exploration scheme that satisfies these restrictions.

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