3.2. Assigning Tasks Suppose a data processing department wishes to assign five programmers to five programming tasks (one programmer to each task). Management has estimated the total number of days each programmer would take if assigned to the different jobs, and these estimates are summarized in the following table.

Task 1 2 3 4 5

1 50 25 78 64 60

2 43 30 70 56 72

Programmer 3 60 28 80 66 68

4 54 29 75 60 70

5 45 32 70 62 75

Determine the assignment that minimizes the total programmer days required to complete all five jobs.

Show the network diagram corresponding to the solution in (a). That is, label each of the arcs in the solution and verify that the flows are consistent with the given information.

How would your solution change if programmer 3 could not be assigned to tasks 2 or 4?

3.4. Distributing a Product The Lincoln Lock Company manufactures a commercial security lock at plants in Atlanta, Louisville, Detroit, and Phoenix. The unit cost of production at each plant is \$35.50, \$37.50, \$37.25, and \$36.25, and the annual capacities are 18,000, 15,000, 25,000, and 20,000, respectively. The locks are sold through wholesale distributors in seven locations around the country. The unit shipping cost for each plant-distributor combination is shown in the following table, along with the forecasted demand from each distributor for the coming year.

Tacoma SanDiego Dallas Denver St Louis TampaBaltimore

Atlanta 2.50 2.75 1.75 2.00 2.10 1.80 1.65

Louisville 1.85 1.90 1.50 1.60 1.00 1.90 1.85

Detroit 2.30 2.25 1.85 1.25 1.50 2.25 2.00

Phoenix 1.90 0.90 1.60 1.75 2.00 2.50 2.65

Demand 5.500 11.500 10.500 9.600 15.400 12.500 6.600

Determine the least costly way of shipping locks from plants to distributors.

Show the network diagram corresponding to the solution in (a). Thatis, label eachof the arcs in the solution and verify that the flows are consistent with the given information.

Suppose that the unit cost at each plant were \$10 higher than the original figure. What change in the optimal distribution plan would result? What general conclusions can you draw for transportation models with nonidentical plant-related costs?

3.8. College Expenses Revisited Revisit the college expense planning network example of this chapter. Suppose the rates on the four investments A, B, C, and D have dropped to 5, 11, 18, and 55 percent, respectively. Suppose that the estimated yearly costs of college (in thousands) have been revised to 25, 27, 30, and 33.

What is the minimum investment that will cover these expenses?

Show the network diagram corresponding to the solution in (a). That is, label each of the arcs in the solution and verify that the flows are consistent with the given information.