During the next four months, a construction firm must complete three projects. Project 1 must be completed within three months and requires 8 months of labor. Project 2 must be completed within four months and requires 10 months of labor. Project 3 must be completed at the end of two months and requires 12 months of labor. Each month, 8 workers are available. During a given month, no more than 6 workers can work on a single job.
a) Draw the network for the maximum flow problem that can be used to determine whether all tasks can be assigned to a suitable worker. (Specify the source and sink nodes, arc direction, and arc capacity).
b) Formulate a maximum-flow problem as a linear programming problem (Define all variables and constraints).
c) Solve the model using the Ford-Fulkerson algorithm and show all iterations. Can all 3 projects be completed in time?
d) Use the final iteration of the Ford-Fulkerson algorithm to find a minimum-cut for this network and verify the strong duality (i.e., maximum flow = total capacity of the minimum cut)