Reconsider the Tinker Construction Co. problem presented in Problem 16.17. While in college, Sean Murphy took a management science course that devoted a month to linear programming, so Sean has decided to use linear programming to analyze this problem. a. Consider the upper path through the project network. Formulate a two-variable linear programming model (in algebraic form) for the problem of how to minimize the cost of performing this sequence of activities within 12 months. Use the graphical method to solve this model. b. Repeat part a for the lower path through the project network. c. Combine the models in parts a and b into a single complete linear programming model (in algebraic form) for the problem of how to minimize the cost of completing the project within 12 months. What must an optimal solution for this model be? E* d. Formulate and solve a spreadsheet model in the format of Figure 16.14 for this problem. E* e. Check the effect of changing the deadline by re-solving this model with a deadline of 11 months and then with a deadline of 13 months. 16.19. Reconsider the Electronic Toys Co. problem presented in Problem 16.14. Sharon Lowe is concerned that there is a significant chance that the vitally important deadline of 57 days will not be met. Therefore, to make it virtually certain that the deadline will be met, she has decided to crash the project, using the CPM method of timecost trade-offs to determine how to do this in the most economical way. Sharon now has gathered the data needed to apply this method, as given below.

The normal times are the estimates of the means obtained from the original data in Problem 16.14. The mean critical path gives an estimate that the project will finish in 51 days. However, Sharon knows from the earlier analysis that some of the pessimistic estimates are far larger than the means, so the project duration might be considerably longer than 51 days. Therefore, to better ensure that the project will finish within 57 days, she has decided to require that the estimated project duration based on means (as used throughout the CPM analysis) must not exceed 47 days. a. Consider the lower path through the project network. Use marginal cost analysis to determine the most economical way of reducing the length of this path to 47 days. b. Repeat part a for the upper path through the project network. What is the total crashing cost for the optimal way of decreasing estimated project duration to 47 days? E* c. Formulate and solve a spreadsheet model that fits linear programming for this problem. 16.20.* Good Homes Construction Company is about to begin the construction of a large new home. The companys president, Michael Dean, is currently planning the schedule for this project. Michael has identified the five major activities (labeled A, B, . . ., E) that will need to be performed according to the following project network. He also has gathered the following data about the normal point and crash point for each of these activities. These costs reflect the companys direct costs for the material, equipment, and direct labor required to perform the activities. In addition, the company incurs indirect project costs such as supervision and other customary overhead costs, interest charges for capital tied up, and so forth. Michael estimates that these indirect costs run $5,000 per week. He wants to minimize the overall cost of the project. Therefore, to save some of these indirect costs, Michael concludes that he should shorten the project by doing some crashing to the extent that the crashing cost for each additional week saved is less than $5,000. a. To prepare for analyzing the effect of crashing, find the earliest times, latest times, and slack for each activity when they are done in the normal way. Also identify the corresponding critical path(s) and project duration. b. Use marginal cost analysis to determine which activities should be crashed and by how much to minimize the overall cost of the project. Under this plan, what is the duration and cost of each activity? How much money is saved by doing this crashing? E* c. Now formulate a spreadsheet model that fits linear programming and repeatedly solve it to do part b by shortening the deadline one week at a time from the project duration found in part a.