Problems

[Page 367 ( continued )]

1.

Construct a Gantt chart for the following set of activities and indicate the project completion time:

Activity	Activity Predecessor	Time (weeks)
1		5
2		4
3	1	3
4	2	6

2.

Construct a Gantt chart for the following set of activities and indicate the project completion time and slack for each activity:

---

[Page 368]

Activity	Activity Predecessor	Time (weeks)
1		3
2		7
3	1	2
4	2	5
5	2	6
6	4	1
7	5	4

3.

Construct a Gantt chart and project network for the following set of activities, compute the length of each path in the network, and indicate the critical path :

Activity	Activity Predecessor	Time (months)
1		4
2		7
3	1	8
4	1	3
5	2	9
6	3	5
7	3	2
8	4, 5, 6	6
9	2	5

4.

Identify all the paths in the following network, compute the length of each, and indicate the critical path (activity times are in weeks):

5.

For the network in Problem 4, determine the earliest and latest activity times and the slack for each activity. Indicate how the critical path would be determined from this information.

6.

Given the following network, with activity times in months, determine the earliest and latest activity times and slack for each activity. Indicate the critical path and the project duration.

---

[Page 369]

---

[Page 370]

7.

Given the following network, with activity times in weeks, determine the earliest and latest activity times and the slack for each activity. Indicate the critical path and the project duration:

---

[Page 371]

8.

In one of the little-known battles of the Civil War, General Tecumseh Beauregard lost the Third Battle of Bull Run because his preparations were not complete when the enemy attacked . If the critical path method had been available, the general could have planned better. Suppose that the following planning network, with activity times in days, had been available:

Determine the earliest and latest activity times and the activity slack for the network. Indicate the critical path and the time between the general's receipt of battle orders and the onset of battle.

9.

A group of developers is building a new shopping center. A consultant for the developers has constructed the following project network and assigned activity times, in weeks.

---

[Page 372]

---

[Page 373]

Determine the earliest and latest activity times, activity slack, critical path, and duration for the project.

10.

A farm owner is going to erect a maintenance building with a connecting electrical generator and water tank. The activities, activity descriptions, and estimated durations are given in the following table:

Activity	Activity Description	Activity Predecessor	Activity Duration (weeks)
a	Excavate		2
b	Erect building	a	6
c	Install generator	a	4
d	Install tank	a	2
e	Install maintenance equipment	b	4
f	Connect generator and tank to building	b, c, d	5
g	Paint finish	b	3
h	Check out facility	e, f	2

(Notice that the activities are defined not by node numbers but by activity descriptions. This alternative form of expressing activities and precedence relationships is often used in CPM/PERT.) Construct the network for this project, identify the critical path, and determine the project duration time.

11.

Given the following network and activity time estimates, determine the expected time and variance for each activity and indicate the critical path:

---

[Page 374]

12.

The Farmer's American Bank of Leesburg is planning to install a new computerized accounts system. Bank management has determined the activities required to complete the project, the precedence relationships of the activities, and activity time estimates, as shown in the following table:

			Time Estimates (weeks)
Activity	Activity Description	Activity Predecessor	a	m	b
a	Position recruiting		5	8	17
b	System development		3	12	15
c	System training	a	4	7	10
d	Equipment training	a	5	8	23
e	Manual system test	b, c	1	1	1
f	Preliminary system changeover	b, c	1	4	13
g	Computerpersonnel interface	d, e	3	6	9
h	Equipment modification	d, e	1	2.5	7
i	Equipment testing	h	1	1	1
j	System debugging and installation	f, g	2	2	2
k	Equipment changeover	g, i	5	8	11

Determine the expected project completion time and variance and determine the probability that the project will be completed in 40 weeks or less.

13.

The following activity time estimates are for the network in Problem 6:

	Time Estimates (mo.)
Activity	a	m	b
1	4	8	12
2	6	10	15
3	2	10	14
4	1	4	13
5	3	6	9
6	3	6	18
7	2	8	12
8	9	15	22
9	5	12	21
10	7	20	25
11	5	6	12
12	3	8	20

Determine the following:

1. Expected activity times

2. Earliest activity times

3. Latest activity times

4. Activity slack

5. Critical path

6. Expected project duration and variance

---

[Page 375]

14.

The following activity time estimates are for the network in Problem 8:

	Time Estimates (days)
Activity	a	m	b
1	1	2	6
2	1	3	5
3	3	5	10
4	3	6	14
5	2	4	9
6	2	3	7
7	1	1.5	2
8	1	3	5
9	1	1	5
10	2	4	9
11	1	2	3
12	1	1	1

Determine the following:

1. Expected activity times

2. Earliest activity times

3. Latest activity times

4. Activity slack

5. Critical path

6. Expected project duration and variance

15.

For the CPM/PERT network in Problem 13, determine the probability that the network duration will exceed 50 months.

16.

The Center for Information Technology at State University has outgrown its office in Bates (B) Hall and is moving to Allen (A) Hall, which has more space. The move will take place during the 3-week break between the end of summer semester and the beginning of fall semester. Movers will be hired from the university's physical plant to move the furniture, boxes of books, and files that the faculty will pack. The center has hired a local retail computer firm to move its office computers so they won't be damaged. Following is a list of activities, their precedence relationships, and probabilistic time estimates for this project:

			Time Estimates (days)
Activity	Activity Description	Activity Predecessor	a	m	b
a	Pack A offices		1	3	5
b	Network A offices		2	3	5
c	Pack B offices		2	4	7
d	Movers move A offices	a	1	3	4
e	Paint and clean A offices	d	2	5	8
f	Move computers	b, e	1	2	2
g	Movers move B offices	b, c, e	3	6	8
h	Computer installation	f	2	4	5
i	Faculty move and unpack	g	3	4	6
j	Faculty set up computers and offices	h, i	1	2	4

Determine the earliest and latest start and finish times, the critical path, and the expected project duration. What is the probability that the center will complete its move before the start of the fall semester?

---

[Page 376]

17.

Jane and Jim Smith are going to give a dinner party on Friday evening at 7:00 P.M . Their two children, Jerry and Judy, are going to help them get ready. The Smiths will all get home from work and school at 4:00 P.M. Jane and Jim have developed a project network to help them schedule their dinner preparations. Following is a list of the activities, the precedence relationships, and the activity times involved in the project:

			Time Estimates (min.)
Activity	Activity Description	Activity Predecessor	a	m	b
a	Prepare salad		18	25	31
b	Prepare appetizer		15	23	30
c	Dust/clean		25	40	56
d	Vacuum	c	20	30	45
e	Prepare dessert	a, b	12	21	30
f	Set table	d	10	17	25
g	Get ice from market	f	5	12	18
h	Prepare/start tenderloin	e	10	20	25
i	Cut and prepare vegetables	e	9	15	22
j	Jim shower and dress	g	8	10	15
k	Jane shower and dress	h, i, j	20	27	40
l	Uncork wine/water carafes	h, i, j	6	10	15
m	Prepare bread	k, l	4	7	10
n	Prepare and set out dishes	k, l	10	14	20
o	Cut meat	k, l	7	15	20
p	Heat vegetable dish	k, l	3	4	6
q	Put out appetizers	m	4	6	7
r	Pour champagne	o	5	8	10

Develop a project network and determine the earliest and latest start and finish times, activity slack, and critical path. Compute the probability that they will be ready by 7:00 P.M.

18.

The Stone River Textile Mill was inspected by OSHA and found to be in violation of a number of safety regulations. The OSHA inspectors ordered the mill to alter some existing machinery to make it safer (add safety guards , etc.); purchase some new machinery to replace older, dangerous machinery; and relocate some machinery to make safer passages and unobstructed entrances and exits. OSHA gave the mill only 35 weeks to make the changes; if the changes were not made by then, the mill would be fined $300,000.

---

[Page 377]

The mill determined the activities in a CPM/PERT network that would have to be completed and then estimated the indicated activity times, as shown in the following table:

			Time Estimates (weeks)
Activity	Activity Description	Activity Predecessor	a	m	b
a	Order new machinery		1	2	3
b	Plan new physical layout		2	5	8
c	Determine safety changes in existing machinery		1	3	5
d	Receive equipment	a	4	10	25
e	Hire new employees	a	3	7	12
f	Make plant alterations	b	10	15	25
g	Make changes in existing machinery	c	5	9	14
h	Train new employees	d, e	2	3	7
i	Install new machinery	d, e, f	1	4	6
j	Relocate old machinery	d, e, f, g	2	5	10
k	Conduct employee safety orientation	h, i, j	2	2	2

Construct the project network for this project and determine the following:

1. Expected activity times

2. Earliest and latest activity times and activity slack

3. Critical path

4. Expected project duration and variance

5. The probability that the mill will be fined $300,000

19.

In the Third Battle of Bull Run, for which a CPM/PERT network was developed in Problem 14, General Beauregard would have won if his preparations had been completed in 15 days. What would the probability of General Beauregard's winning the battle have been?

20.

On May 21, 1927, Charles Lindbergh landed at Le Bourget Field in Paris, completing his famous transatlantic solo flight. The preparation period prior to his flight was quite hectic, and time was very critical because several other famous pilots of the day were also planning transatlantic flights . Once Ryan Aircraft was contracted to build the Spirit of St. Louis , it took only a little over 2.5 months to construct the plane and fly it to New York for the takeoff. If CPM/PERT had been available to Charles Lindbergh, it no doubt would have been useful in helping him plan this project. Use your imagination and assume that a CPM/PERT network with the following estimated activity times was developed for the flight.

---

[Page 378]

---

[Page 379]

	Time Estimates (days)
Activity	a	m	b
1	1	3	5
2	4	6	10
3	20	35	50
4	4	7	12
5	2	3	5
6	8	12	25
7	10	16	21
8	5	9	15
9	1	2	2
10	6	8	14
11	5	8	12
12	5	10	15
13	4	7	10
14	5	7	12
15	5	9	20
16	1	3	7

Determine the expected project duration and variance and the probability of completing the project in 67 days.

21.

RusTech Tooling is a large job shop operation that builds machine tools and dies to manufacture parts for specialized items. The company primarily bids on government- related contracts to produce parts for such items as military aircraft and weapons systems, as well as the space program. The company is bidding on a contract to produce a component part for the fuselage assembly in a new space shuttle. A major criterion for selecting the winning bid, besides low cost, is the time required to produce the part. However, if the company is awarded the contract, it will be strictly held to the completion date specified in the bid, and any delays will result in severe financial penalties. To determine the project completion time to put in its bid, the company has identified the project activities, precedence relationships, and activity times shown in the following table:

		Time Estimates (weeks)
Activity	Activity Predecessor	a	m	b
a		3	5	9
b	a	2	5	8
c	a	1	4	6
d	a	4	6	10
e	b	2	8	11
f	b	5	9	16
g	c	4	12	20
h	c	6	9	13
i	d	3	7	14
j	d	8	14	22
k	f, g	9	12	20
l	h, i	6	11	15
m	e	4	7	12
n	j	3	8	16
o	n	5	10	18

---

[Page 380]

If RusTech wants to be 90% certain that it can deliver the part without incurring a penalty, what time frame should it specify in the bid?

22.

PM Computers is an international manufacturer of computer equipment and software. It is going to introduce a number of new products in the coming year, and it wants to develop marketing programs to accompany the product introductions . The marketing program includes the preparation of printed materials distributed directly by the company and used by the company's marketing personnel, vendors, and representatives; print advertising in regular magazines, trade journals, and newspapers; and television commercials. The program also includes extensive training programs for marketing personnel, vendors , and representatives about the new products. A project management team with members from the marketing department and manufacturing areas has developed the following list of activities for the development of the marketing program:

			Time Estimates (days)
Activity	Activity Description	Activity Predecessor	a	m	b
a	Preliminary budget and plan approval		10	15	20
b	Select marketing personnel for training		5	9	12
c	Develop overall media plan	a	15	25	30
d	Prepare separate media plans	c	12	20	25
e	Develop training plan	c	5	8	12
f	Design training course	e	6	14	20
g	Train marketing personnel	b, f	16	20	25
h	Plan TV commercials with agency	d	15	25	35
i	Draft in-house print materials	d	8	15	20
j	Develop print advertising layouts with agency	d	16	23	30
k	Review print advertising layouts	j	4	9	12
l	Review TV commercials	h	3	7	12
m	Review and print in-house materials	i	3	5	7
n	Release advertising to print media	g, i, k	2	4	8
o	Release TV commercials to networks	l	4	7	10
p	Final marketing personnel review	g, i, k	4	5	9
q	Run media advertising, mailings	m, n, o	15	20	30

Construct the network for this project and determine the activity schedule. Identify the critical path and determine the expected project duration time and variance. What is the probability that the program can be completed within 4 months?

23.

A marketing firm is planning to conduct a survey of a segment of the potential product audience for one of its customers. The planning process for preparing to conduct the survey consists of six activities, with precedence relationships and activity time estimates as follows :

Activity	Activity Description	Activity Predecessor	Time Estimate (days)
a	Determine survey objectives		3
b	Select and hire personnel	a	3
c	Design questionnaire	a	5
d	Train personnel	b, c	4
e	Select target audience	c	3
f	Make personnel assignments	d, e	2

1. ---

   [Page 381]

2. Determine all paths through the network from node a to node f and the duration of each. Also indicate the critical path.

3. Determine the earliest and latest activity start and finish times.

4. Determine the slack for each activity.

24.

Lakeland-Bering Aircraft Company is preparing a contract proposal to submit to the defense department for a new military aircraft, the X-300J jet fighter. Part of the proposal is a development and productions schedule for completion of the first aircraft. The project consists of three primary categories: engine design and development, development and production of the airframe (e.g., the aircraft body), and design and development of the aircraft avionics (e.g., the electronic systems, equipment, and other devices used to operate and control the aircraft). Following is a list of the project activities, with time estimates (in months):

			Time Estimates (mo.)
Activity	Activity Description	Activity Predecessor	a	m	b
1	General design		5	10	22
2	Engine design		20	33	57
3	Airframe design	1	6	17	33
4	Avionics design	1	2	8	20
5	Develop test engine	2	7	10	12
6	Develop test airframe	3	8	11	18
7	Develop interim avionics	4	7	15	28
8	Develop engine	2	19	24	41
9	Assemble test aircraft	5, 6, 7	4	6	10
10	Test avionics	4	5	10	16
11	Conduct engine/airframe flight trials	9	5	9	12
12	Conduct avionics flight trials	10	5	9	24
13	Produce engine	8	11	14	15
14	Produce airframe	11	12	15	20
15	Produce avionics	11, 12	14	16	25
16	Final assembly/finish	13, 14, 15	2	3	5

Develop the project network and determine the critical path, the expected project duration, and the variance. What is the probability that the project will be completed within 8 years ?

25.

The Valley United Soccer Club is planning a soccer tournament for the weekend of April 29 and 30. The club's officers know that by March 30 they must send out acceptances to teams that have applied to enter and that by April 15 they must send out the tournament game schedule to teams that have been selected to play. Their tentative plan is to begin the initial activities for tournament preparation, including sending out the application forms to prospective teams, on January 20. Following is a list of tournament activities, their precedence relationships, and estimates of their duration, in days:

---

[Page 382]

			Time Estimates (days)
Activity	Activity Description	Activity Predecessor	a	m	b
a	Send application forms		5	7	10
b	Get volunteer workers		10	18	26
c	Line up referees		7	10	14
d	Reserve fields		14	21	35
e	Receive and process forms	a	30	30	30
f	Determine age divisions	b, c, d, e	4	9	12
g	Assign fields to divisions	f	4	7	10
h	Sell program ads	b	14	21	30
i	Acquire donated items for team gift bags	b	15	20	26
j	Schedule games	g	6	14	18
k	Design ads	h	5	8	10
l	Fill gift bags	i	9	12	17
m	Process team T-shirt orders	e	7	10	14
n	Send acceptance letters	f	4	7	12
o	Design and print programs	j, k, l, n	14	18	24
p	Put together registration boxes (programs, gifts, etc.)	o	5	7	10
q	Send out game schedules	j, k, l, n	5	8	12
r	Assign referees to games	j, k, l, n	4	7	10
s	Get trophies	f	20	28	35
t	Silk-screen T-shirts	m	12	17	25
u	Package team T-shirt orders	t	5	8	12

Develop a project network for the club's tournament preparation process and determine the likelihood that it will meet its schedule milestones and complete the process according to the scheduled tournament date of April 29.

26.

During a violent thunderstorm with very high wind gusts in the third week of March, the broadcast tower for the public radio station WVPR, atop Poor Mountain in Roanoke, collapsed . This greatly reduced the strength of the station's signal in the area. The station management immediately began plans to construct a new tower. Following is a list of the required activities for building the new tower with optimistic ( a ), most likely ( m ), and pessimistic ( b ) time estimates (in days); however, the sequence of the activities has not been designated:

		Time Estimates (days)
Activity	Activity Description	a	m	b
a	Removal of debris	5	8	12
b	Survey new tower site	3	6	8
c	Procure structural steel	15	21	30
d	Procure electrical/broadcasting equipment	18	32	40
e	Grade tower site	4	7	10
f	Pour concrete footings and anchors	10	18	22
g	Deliver and unload steel	3	5	9
h	Deliver and unload electrical/broadcast equipment	1	2	4
i	Erect tower	25	35	50
j	Connect electrical cables between tower and building	4	6	10
k	Construct storm drains and tiles	10	15	21
l	Backfill and grade tower site	4	7	9
m	Clean up	3	6	10
n	Obtain inspection approval	1	4	7

---

[Page 383]

Using your best judgment, develop a CPM/PERT network for this project and determine the expected project completion time. Also determine the probability that the station signal will be back at full strength within 3 months.

27.

The following table contains the activities for planning a wedding and the activity time estimates; however, the precedence relationships between activities are not included:

		Time (days)
Activity	Activity Description	a	m	b
a	Determine date	1	10	15
b	Obtain marriage license	1	5	8
c	Select bridal attendants	3	5	7
d	Order dresses	10	14	21
e	Fit dresses	5	10	12
f	Select groomsmen	1	2	4
g	Order tuxedos	3	5	7
h	Find and rent church	6	14	20
i	Hire florist	3	6	10
j	Develop/print programs	15	22	30
k	Hire photographer	3	10	15
l	Develop guest list	14	25	40
m	Order invitations	7	12	20
n	Address and mail invitations	10	15	25
o	Compile RSVP list	30	45	60
p	Reserve reception hall	3	7	10
q	Hire caterer	2	5	8
r	Determine reception menu	10	12	16
s	Make final order	2	4	7
t	Hire band	10	18	21
u	Decorate reception hall	1	2	3
v	Wedding ceremony	0.5	0.5	0.5
w	Wedding reception	0.5	0.5	0.5

Using your best judgment, determine the project network, critical path, and expected project duration. If it is January 1 and a couple is planning a June 1 wedding, what is the probability that it can be done on time?

28.

The following table provides the information necessary to construct a project network and project crash data:

			Activity Time (weeks)			Activity Cost
Activity	( i , j )	Activity Predecessor	Normal	Crash		Normal	Crash
a	(1, 2)		20	8		$1,000	$1,480
b	(1, 4)		24	20		1,200	1,400
c	(1, 3)		14	7		700	1,190
d	(2, 4)	a	10	6		500	820
e	(3, 4)	c	11	5		550	730

1. Construct the project network.

2. Compute the total allowable crash time per activity and the crash cost per week for each activity.

   ---

   [Page 384]

3. Determine the maximum possible crash time for the network and manually crash the network the maximum amount.

4. Compute the normal project cost and the cost of the crashed project.

5. Formulate the general linear programming model for this network and solve it.

6. Formulate the linear programming crashing model that would crash this network by the maximum amount and solve it.

29.

The following table provides the information necessary to construct a project network and project crash data:

			Activity Time (weeks)			Activity Cost
Activity	( i , j )	Activity Predecessor	Normal	Crash		Normal	Crash
a	(1, 2)		16	8		$2,000	$4,400
b	(1, 3)		14	9		1,000	1,800
c	(2, 4)	a	8	6		500	700
d	(2, 5)	a	5	4		600	1,300
e	(3, 5)	b	4	2		1,500	3,000
f	(3, 6)	b	6	4		800	1,600
g	(4, 6)	c	10	7		3,000	4,500
h	(5, 6)	d, e	15	10		5,000	8,000

1. Construct the project network.

2. Manually crash the network to 28 weeks.

3. Formulate the general linear programming model for this network.

4. Formulate the linear programming crashing model that would crash this model by the maximum amount.

30.

Formulate the general linear programming model for Problem 4, and solve it.

31.

Formulate the general linear programming model for the project network for installing an order processing system shown in Figure 8.14 and solve it.

32.

Reconstruct the example problem at the end of this chapter as an AOA network. Assume that the most likely times ( m ) are the normal activity times and that the optimistic times ( a ) are the activity crash times. Further assume that the activities have the following normal and crash costs:

Activity	( i , j )	Costs (normal cost, crash cost)
1	(1, 2)	($100, $400)
2	(1, 3)	($250, $400)
3	(2, 3)	($400, $800)
4	(2, 4)	($200, $400)
5	(3, 4)	($150, $300)
6	(3, 5)	($100, $100)
7	(4, 5)	($300, $500)

1. Formulate the general linear programming model for this project network, using expected activity times ( t ).

   ---

   [Page 385]

2. Formulate the linear programming crashing model that would crash this network by the maximum amount.

3. Solve this model by using the computer.

33.

The following table provides the crash data for the network project described in Problem 12:

	Activity Time (weeks)			Activity Cost
Activity	Normal	Crash		Normal	Crash
a	9	7		$4,800	$ 6,300
b	11	9		9,100	15,500
c	7	5		3,000	4,000
d	10	8		3,600	5,000
e	1	1
f	5	3		1,500	2,000
g	6	5		1,800	2,000
h	3	3
i	1	1
j	2	2
k	8	6		5,000	7,000

The normal activity times are considered to be deterministic and not probabilistic. Using the computer, crash the network to 26 weeks. Indicate how much it would cost the bank and then indicate the critical path.

34.

The following table provides the crash data for the network project described in Problem 6:

	Activity Time (months)			Activity Cost ($1,000s)
Activity	Normal	Crash		Normal	Crash
1	8	5		$ 700	$1,200
2	10	9		1,600	2,000
3	9	7		900	1,500
4	4	2		500	700
5	6	3		500	900
6	5	4		500	800
7	7	5		700	1,000
8	15	12		1,400	2,000
9	12	10		1,800	2,300
10	18	14		1,400	3,200
11	4	3		500	800
12	7	6		800	1,400

Using the computer, crash the network to 32 months. Indicate the first critical path activities and then the cost of crashing the network.