OPERATIONS RESEARCH
Exercises
Project Management
运筹学课业代做 Ex. 7Consider the project of exercise 6. Suppose the project started 8 weeks ago and: activities A and G have already been completed,
Ex. 1
Draw the AOA and AON networks representing the following projects:
a)
| Activities | Duration | Immediate predecessors |
| A | 4 | — |
| B | 8 | A |
| C | 9 | A |
| D | 7 | B,C |
| E | 5 | D |
b)
| Activities | Duration | Immediate predecessors |
| A | 4 | — |
| B | 2 | — |
| C | 3 | E |
| D | 2 | A |
| E | 1 | B,D |
| F | 4 | D |
| G | 5 | F |
c)
| Activities | Duration | Immediate predecessors |
| A | 8 | — |
| B | 9 | — |
| C | 6 | A |
| D | 5 | B |
| E | 4 | A, B |
| F | 3 | C, E |
| G | 7 | C, D, E |
Ex. 2 运筹学课业代做
Consider the following project:
| ACTIVITIES | PREDECESSORS | DURATION |
| A | – | 10 |
| B | – | 5 |
| C | A | 10 |
| D | B | 10 |
| E | A, B | 15 |
| F | C | 12 |
| G | C, E | 15 |
| H | D | 10 |
a) Draw the AON network for the project.
b) Determine the minimum project duration. Which activities are critical?
Ex. 3
Consider the following project:
a) Draw the AON network for the project.
b) Which activities are critical?
c) Is it realistic to consider a deadline of 20 weeks to conclude the project?
Ex. 5
Consider the following project:
| ACTIVITIES | ACTIVITIES | DURATION (in days) |
| A | – | 2 |
| B | – | 3 |
| C | A | 3 |
| D | A, B | 4 |
| E | B | 3 |
| F | C, D | 3 |
| G | D, E | 2 |
| H | D, E | 3 |
| I | F, G | 2 |
| J | G, H | 4 |
a) Draw the AON network for the project.
b) Determine the minimum project duration. Which activities are critical?
Ex. 6 运筹学课业代做
Consider the project represented in the following network.
| Activities | A | B | C | D | E | F | G | H | I | J | K | L | M |
| Durations | 6 | 4 | 7 | 2 | 4 | 10 | 2 | 10 | 13 | 6 | 9 | 3 | 5 |
a) Determine the minimum project duration, and identify the critical path.
b) Analyze the effect on the minimum project duration, on the critical activities and on the critical path if the duration of activity K is increased by 1week.
Ex. 7
Consider the project of exercise 6. Suppose the project started 8 weeks ago and: activities A and G have already been completed, B is going to be completed in 3 weeks, D is going to be completed in 1 week, and I is going to be completed in 10 weeks. The remaining activities have not started yet. Will the minimum project duration found in Ex. 6 change? What will be the new minimum project duration?
Ex. 8 运筹学课业代做
Consider the project represented in the following AON network:
In the following table, the duration and the resources required to complete activities as well as the earliest and latest start times are given:
| Activities | A | B | C | D | E | F | G | H |
| Durations | 4 | 3 | 1 | 4 | 5 | 2 | 4 | 2 |
| Resources | 2 | 3 | 2 | 2 | 3 | 2 | 4 | 2 |
| Earliest Start Time | 0 | 0 | 4 | 4 | 4 | 8 | 9 | 9 |
| Latest Start Time | 0 | 1 | 8 | 5 | 4 | 11 | 9 | 11 |
a) Find SE, draw the corresponding Gantt chart and Resource Allocation Histogram.
b) Find SL, draw the corresponding Gantt chart and Resource Allocation Histogram.
c) Compare SEand SL.
d) Consider now the schedule S = (4, 3, 5, 9, 9, 11, 13, 13). Draw the Gantt chart. Is this schedule better than SE? And than SL?
Ex. 9 运筹学课业代做
(Adapted from an exam of OR II OGE 2002/03) Consider a project made up of activities whose duration (in weeks) may be reduced. The available reduction as well as the marginal cost for each activity is given in the following table:
| Activity | A | B | C | D | E | F | G | H |
| Available Reduction | 2 | 3 | 1 | 1 | 2 | 3 | 3 | 1 |
| Marginal Cost | 60 | 150 | 75 | 120 | 55 | 80 | 65 | 45 |
The possible paths (in the AON network) between the start and the finish of the project and their durations (in weeks) are listed in following table:
| Path | Duration |
| A-D-G | 58 |
| A-F | 27 |
| B-G | 50 |
| C-E-G | 60 |
| C-H | 36 |
After applying two iterations of the algorithm to reduce the minimum duration of the project the following results were obtained.
| Iteration | Reduced Activities | Reduction | Duration of the project | Marginal Cost | Increase of Cost | Cost of the Project | New Critical Path |
| 0 | – | – | 60 | – | – | 1635 | C-E-G |
| 1 | E | 2 | 58 | 55 | 110 | 4745 | A-D-G |
| 2 | G | 3 | 55 | 65 | 195 | 4940 | – |
a) Rewrite the Marginal Cost table.
b) Rewrite the Possible Paths table.
c) Finish the application of the algorithm.
Ex. 10
(Adapted from an exam of OR II OGE 2002/03) Consider the following AON network representing a project with 5 activities.
The duration and the resources required to complete activities are given in the following table:
| Activities | A | B | C | D | E |
| Durations | 20 | 25 | 50 | 40 | 30 |
| Resources | 5 | 10 | 20 | 5 | 15 |
a) Find the schedule SE.
b) Draw the Resource Allocation Histogram for SL. What is the maximum number of resources needed in this schedule?
c) The maximum number of resources available (every week) is 25. Apply the SGS – P heuristic in order to obtain a resource feasible schedule. Consider the shorter duration as priority rule.
Ex. 11
(Adapted from an exam of exam of OR II GEI 2002/03) Consider the following project.
The normal duration, the available reduction and the marginal cost for each activity are given in the following table:
| Activity | A | B | C | D | E | F | G |
| Normal Duration | 5 | 5 | 8 | 4 | 3 | 6 | 3 |
| Available Reduction | 2 | 1 | 1 | 1 | 2 | 3 | 2 |
| Marginal Cost | 60 | 60 | 75 | 120 | 55 | 80 | 45 |
The possible paths (in the AON network) between the start and the finish of the project are listed in following table:
| Paths | Duration |
| A-D-G | |
| B-C | |
| B-E-F |
a) Complete the Possible Paths Table.
b) What will be the minimum project during, after applying the algorithm? Do not forget to fill in and update the tables used to apply the algorithm.
Ex. 12 运筹学课业代做
(Adapted from an exam of Exam OR II GEI 2002/03) Consider the project:
and the following related information:
| Activity | Normal Duration (weeks) | Normal Cost | Crashed Duration (weeks) | Crashed Cost | Resources |
| A | 2 | 500 | 1 | 800 | 8 |
| B | 5 | 900 | 3 | 1080 | 5 |
| C | 4 | 800 | 3 | 1000 | 2 |
| D | 1 | 400 | 1 | 400 | 4 |
| E | 3 | 1200 | 2 | 1800 | 6 |
| F | 6 | 700 | 4 | 900 | 5 |
| G | 8 | 600 | 4 | 1200 | 3 |
a) Determine the minimum duration of the project and the critical path.
b) Is it possible to reduce the minimum project duration to 10 weeks? How? What will be the cost?
c) Suppose there are 14 units of resource available. Find a resource-feasible schedule using the appropriate algorithm. Consider as priority rule the maximum duration and to break ties the smallest Total Float. Is there any delay in comparison to SE?
d) Suppose again that there are 14 units of resource available. Find aresource-feasible schedule using the appropriate algorithm. Consider as priority rule the minimum duration and to break ties the smallest Total Float. Is there any delay in comparison to SE?
Ex 13 运筹学课业代做
Consider the following table containing the information related to a project which has to be completed in 14 days:
| Activity | Predecessors | Normal duration (in days) | Resources needed per day | Normal cost | Crash Duration (in days) | Crash cost |
| A | — | 2 | 3 | 1000 | 1 | 2000 |
| B | A | 4 | 2 | 2000 | 2 | 2500 |
| C | A | 3 | 4 | 1400 | 2 | 1800 |
| D | B | 3 | 2 | 1500 | 2 | 1800 |
| E | B,C | 2 | 5 | 800 | 1 | 1500 |
| F | E | 4 | 3 | 1600 | 2 | 2400 |
a) Draw the AON network for this project.
b) Find the minimum project duration.
c) Which activities will delay the whole project (will change the minimum duration of the project) if they are delayed?
d) Do the minimum duration of the project, the critical activities and the critical path change if the duration of C increases by 2 days? What are the changes?
e) Reduce the minimum duration of the project until it becomes impossible to decrease it more such that the total direct cost is minimized. To obtain the initial minimum project duration problem consider for each activity its normal duration. Draw the duration-cost curve for this project. 运筹学课业代做
f) Consider the precedence between activities, durations and daily consumption of resources.
i) Find SEand SL.
ii) Draw the Gantt chart for SEand for SL.
iii) Draw the RAH associated with SE and SL. What is the maximum number of resources needed for each schedule?
iv) Which of the schedules is the best?
g) Suppose that there are 5 units of resource available. Apply the SGS-Parallel heuristic.
Consider as priority rule the Total Float, and for breaking ties the shortest duration.
RESUMED SOLUTIONS 运筹学课业代做
NOTE: Solutions may have to be interpreted according to the context of the problem.
Ex 2
b) Minimum Duration of the Project = 40. Critical activities: A, E, G.
Ex 3
b) Critical activities: A, B, C, E, G.
c) No, the probability is close to 0,0002.
Ex 4
a) Pessimist estimate for C =11; Optimist estimate for G = 2
| Activ. | A | B | C | D | E | F | G |
| ES | 0 | 0 | 5 | 5 | 9 | 9 | 12 |
| EF | 5 | 4 | 9 | 12 | 11 | 14 | 18 |
| LS | 0 | 1 | 6 | 5 | 10 | 13 | 12 |
| LF | 5 | 5 | 10 | 12 | 12 | 18 | 18 |
b) Mean Duration of the Project = 18, Variance of Duration of the Project = 70/9.(Critical Activities: A, D, G).
Ex 5 运筹学课业代做
a) (Pay Attention: One pair of arcs is crossed)
b) Minimum Duration of the Project = 14. Critical Activities: B, D, H, J.
Ex 6
a) Minimum Duration of the Project = 25 weeks. Critical Path: A – B – C – L – M.
b) There are no changes;
Ex 7 It is necessary to redraw the network. The minimum duration of the project increases from 25 to 29 weeks.
Ex 8
a) SE = (4, 3, 5, 8, 9, 10, 13, 11)
| Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Resource | 5 | 5 | 5 | 2 | 7 | 5 | 5 | 5 | 5 | 8 | 6 | 4 | 4 |
b) SL= (4, 4, 9, 9, 9, 13, 13, 13)
| Time | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
| Resource | 2 | 5 | 5 | 5 | 3 | 5 | 5 | 5 | 7 | 4 | 4 | 8 | 8 |
b) SEand SLhave equal Utilization Ration and Max Float. The variance is smaller for SE. So SE is better than SL.
c) This schedule has higher Utilization Ratio, smaller Max Float and smaller Variance so it is better than SEand SL.
Ex 9 运筹学课业代做
c) 3rdIteration – dAand dC are decreased in 1 unit of time. Duration of the Project = Cost = 5075.
Ex 10 a) SE = (20, 25, 70, 65, 100)
b) SL= (20, 30, 70, 70, 100). This Schedule requires, at most, 30 units of resource.
c) Schedule = (20, 25, 75, 65, 105). d. p. = 105
Ex 11 b) Minimum Duration = 11. (1st Iteration – dE is decreased in 1 week, 2nd Iteration – dB is decreased in 1 week, 3rd Iteration dG, dC and dE are decreased in 1 week). Increase of the Directed Cost = 290.
Ex 12 a) Minimum Duration of the Project = 13 weeks. Critical Path: C – E – F.
b) Yes, additional costs = 550.
1st Iteration: dF is decreased in 1 week, 2nd Iteration: dC is decreased in 1 week, 3rd Iteration: dF is decreased in 1 week.
c) Schedule = (9, 5, 4, 10, 7, 16, 12). There is a delay of 3 weeks.
d) Schedule = (2, 7, 4, 3, 7, 13, 12). There is no delay.
