1) The Reynolds Construction Company has received a contract to build a water purification plant; managers at Reynolds have identified the following tasks to assist them with the construction of the remote control building.  In addition to identifying the tasks needed for this project, they estimated the duration and direct costs associated with completing each task at its “Normal” point (the duration with the minimum direct costs) and the duration and direct costs associated with the “Crash” point (that is, if the task is maximally compressed).   The managers assume that the marginal compression costs are constant (for example, the marginal compression cost for task B would be ($26,000 – $14,000)/(6 – 4 weeks)  = $6000/wk). 

Normal                                     Crash

Task              Description             Predecessor          Duration(wks)      Cost($)             Duration(wks)    Cost($

A               Procure materials         Start                               3                      5,000                       2                  10,000

B               Prepare site                    Start                               6                    14,000                       4                  26,000

C               Dpt approval                  Start                              2                     2,500                       1                    5,000

D              Prebabricate bldg.            A                                  5                     10,000                      3                  18,000

E              Obtain city approvals       C                                  2                      8,000                       2                    8,000

F              Install Lines                      A                                    7                     11,000                      5                  17,500

G             Erect bldg.                      B,D,E                               4                     10,000                      2                    24,000

                                                                                                          Totals     61,000                                          108,000

If all tasks are performed at their “normal” times as indicated in the table above, the project is estimated to cost Reynolds a total of $61,000 in direct costs and result in an expected project makespan of 12 weeks.  However, managers at Reynolds want to explore the time-cost trade-offs associated with compressing the project and have come to you for assistance.

Specifically, Reynolds’ managers have asked you to do the following.  Starting with all tasks performed at their normal duration (such that the project makespan is 12 weeks), reduce the makespan of the project by one week at a time.  For each iteration (e.g., 11 weeks, 10 weeks, etc), find the task(s) that should be compressed to minimize the marginal increase in direct costs at each iteration. Continue compressing the project until at least one critical path exists with all tasks set at their crash times (that is, no further project compression is possible).  Construct a graph showing the total direct cost (on the Y axis) versus the project makespan (on the X axis).  Clearly indicate the minimum possible project makespan.

2) The managers at Reynolds Construction Company now realize that their calculations in problem #1 do not include overhead/indirect costs that they estimate at $5100/week (this cost includes the rental fee for security fencing, an overhead construction crane needed for the duration of this project, and your services as manager of this construction project).   They also have signed a contract that calls for a penalty cost of $3200 per week for each week that the makespan exceeds ten weeks.  Given these costs, formulate and solve a linear programming model that finds the baseline schedule that minimizes the total costs of this project (where total costs are the sum of direct, indirect/overhead, and penalty costs).