1
Centrality Inspired Quality Measures for Network Based Schedules
2
Mohsin K. Siddiqui A.M. ASCE1 and Muhammad A. Khan2
3
Abstract:
4
Existing measures for schedule performance are reflective in nature and consider goodness in
5
terms of hindsight assessment of conformance to a baseline. A few predictive measures reported
6
in literature use complexity of a schedule as an indicator of its quality under the premise that
7
more complex schedules would require additional coordination efforts. However, these measures
8
are local and focus on the network topology alone. A predictive set of global quality measures
9
inspired by the concept of centrality from social network analysis are presented in this paper.
10
Unlike the complexity oriented quality measures that are applicable for evaluation of schedule
11
alternatives for a single project, the developed quality measures are applicable for analysis
12
beyond single project schedules and enable comparison between schedules from multiple
13
projects. For this research, quality is defined as the degree to which a schedule can sustain
14
changes. The measures are applicable as long as the changes to a project do not require a
15
significant structural modification of the schedule. A software tool as an MS Project add-in has
16
been developed to use the measures for real project schedules.
17
18
Keywords: Scheduling, Project Management, Construction Industry
19
20
1
Mohsin K. Siddiqui, Ph.D., A.M. ASCE, Assistant Professor, Construction Engineering and
Management, King Fahd University of Petroleum and Minerals, KFUPM Box 1032, Dhahran
31261, Saudi Arabia, [email protected] (corresponding author)
2
Muhammad A. Khan, Lecturer, PYP Mathematics Program, King Fahd University of
Petroleum and Minerals, KFUPM Box 1938, Dhahran 31261, Saudi Arabia,
[email protected]
1
21
1.
Introduction
22
Scheduling of projects becomes a complex task in dynamic and uncertain environments
23
encountered in domains such as construction. A scheduler’s task is compounded partly by the
24
project constraints and partly by the limited understanding of how the schedule will behave
25
under changes. The former challenge is related to the complexity of a project however the
26
behavior of the schedule under changes falls under the quality aspects of a project schedule and
27
is a function of the structure of the schedule. The authors define the quality of a schedule as the
28
extent to which a schedule can withstand changes without requiring considerable alterations
29
(Khan et al. 2010). Other researchers have viewed schedule quality in terms of network
30
complexity alone. Network complexity focuses on interconnection between activities and is
31
presented as an indicator for the effort that would be needed in coordination of the schedule
32
(Nassar et al. 2006) and the time that would spent on planning and scheduling (Badiru et al.
33
1995). However, the behavior under change is not considered in these analyses.
34
35
Changes in project schedules have significant direct costs when key resources are involved in the
36
impacted activities. In addition, hidden costs include the effort to identify the affected
37
participants on large complex projects and the time required to coordinate the intertwined
38
activities of the various execution agents. Therefore, support for a robust response to change is
39
crucial in deciding the viability (and the ultimate use) of project schedules. The problem has
40
received significant attention in different scheduling research domains. Smith (2005), in his
41
broad review of scheduling research in various contexts has considered response to changes
42
among the core challenges facing the scheduling research. Construction is a dynamic process and
43
represents a highly uncertain execution environment. Schedules would typically require
2
44
alteration in response to major scope changes and unforeseen site conditions. Having said that,
45
these cannot be used as excuses for participants on a project to develop impractical schedules;
46
schedules that will break down when perturbed even slightly. From a scheduling perspective, the
47
aforementioned changes in the scope, unfavorable site conditions, and different crew
48
productivity than planned are likely to result in one or more of: (1) increased or decreased
49
activity durations, (2) modification of schedule logic for a small part of the schedule, (3) addition
50
or deletion of some activities, and in the worst case (4) major schedule revisions. The assessment
51
of schedules and the measures presented in this paper are motivated by the first three of the
52
aforementioned scenarios.
53
54
Regardless of the source of the disturbance, there are few tools available that allow owners and
55
other consumers of construction project schedules to assess the quality of a particular schedule or
56
to perform rapid analysis of their options in response to a schedule change. The Monte Carlo
57
based simulation tools (Palisade 2000) are an effective option but these tools require detailed
58
uncertainty information about the tasks and resources; information that is in practice often not
59
available in the schedules. Russell et al. (2000) propose a series of completeness and workability
60
oriented quality measures and visual means of accessing a schedule. However the measures (and
61
their assessment) are subjective in nature, require access to significant information beyond the
62
basic schedule data, and are suited for linear schedules or schedules for repetitive projects. Even
63
if a scheduler has access to such information, the information is not readily available to
64
consumers of the schedules (e.g. owners) who are sometimes responsible for evaluating and
65
approving the schedules.
66
3
67
This paper presents a set of quality measures that can be used to assess schedules using the basic
68
scheduling information – activities, their durations, and the activity precedence relationships.
69
Unlike the complexity measures, these quality measures are global in nature; that is, the
70
measures can be used to compare different schedules for a single project as well as for
71
comparison of schedules from different projects. The paper proceeds with the presentation of two
72
quality measures to enable analysis at the activity level and at the overall schedule level. The
73
first measure, named value centrality, estimates the likelihood of change in the schedule
74
makespan when activity durations are modified. The path centrality, on the other hand, measures
75
the probability of a new critical path being generated in the schedule. The measures are inspired
76
by the concept of betweeness centrality (Freeman 1977) from the social network analysis
77
domain. The upper and lower bounds for these measures are determined using boundary case
78
examples. A software tool has been developed as an MS Project Add-in to enable use of these
79
measures for real project schedules.
80
81
2.
Literature Review
82
2.1
Schedule Uses by Project Stakeholders
83
Project stakeholders generate and use schedules to support higher project management functions.
84
The General Contractor (GC), concerned with the overall project performance, typically prepares
85
a network based Critical Path Method (CPM) master schedule but largely relies on short interval
86
look-ahead schedules for monitoring site activities (Siddiqui et al. 2009). The CPM schedule is
87
used for periodic control of work, developing look-ahead schedules, schedule coordination,
88
detailed planning, and for identifying the impact of changes for claims (Galloway 2006). The
89
subcontractors typically rely on the GC for overall coordination and develop their own schedules
4
90
for monitoring their tasks especially for projects where the master schedule does not fully cover
91
the scope and complexity of the subcontractors’ work (Dossick et al. 2007; Galloway 2006). The
92
owners are mostly consumers of schedules and on most projects would not generate their own
93
schedule for the project. Although the owners report the “what-if analysis” as an advantage of
94
CPM schedules, the owners sometimes feel disenfranchised by the inability of the CPM
95
generated schedules to support their approval, review, and monitoring and control roles
96
(Galloway 2006). The owners sometime tend to believe that the GC can manipulate the CPM
97
schedule to build claims (Galloway 2006; Zack 1992). The owners therefore put in place certain
98
structural requirements to ensure that better quality schedules are generated for their projects
99
(Nassar et al. 2006; Zack 1992). The delay in submission of schedules by the GC also puts
100
owners in a precarious position as they are often ill-equipped to evaluate the complex logic of the
101
schedules in limited time. This challenge of assessment is exacerbated by the scarcity of tools
102
that support reasoning with basic scheduling data.
103
104
2.2
Complexity based Quality Measures
105
106
Several measures have been reported in the literature for quantitative assessment of schedule
107
quality. Most measures approach quality in terms of complexity of the schedule network i.e., a
108
less complex schedule network is considered to have better quality. Generally the coefficients of
109
network complexity (CNC) and their variants (Badiru et al. 1995; Davies 1977; Latva-Koivisto
110
2001) are used as indicators of schedule complexity. However De Reyck et al. (1996) and Nassar
111
et al. (2006) have introduced some new measures. Although these measures interpret complexity
112
in different contexts, in general, a schedule network with more links is considered more
5
113
complex. A complex schedule is considered difficult to interpret and implement and hence a
114
schedule with lesser complexity is regarded as a better schedule (Nassar et al. 2006).
115
Given a project schedule as an Activity on Node (AON) graph G(N,A) with a set N of nodes and
116
A arcs, the measures are given as:
117
Davies ′ Coefficient = 𝐶𝑑 =
118
2 (𝑎 − 𝑛 + 1)
(𝑛 − 1)(𝑛 − 2)
Jhonson′ s Coefficient = 𝐶𝑗 = ∑ max{0, (𝑝𝑥 − 𝑠𝑥 )}
119
𝑥∈𝑁
[
Nassar ′ s Measure = 𝐶𝑛 =
120
log [𝑎⁄(𝑛𝑑 − 1)]
× 100] % if 𝑛𝑑 is odd
(𝑛𝑑 2 − 1)
log [
⁄4(𝑛𝑑 − 1)]
[
{
log [𝑎⁄(𝑛𝑑 − 1)]
(𝑛𝑑 2 )
log [
⁄4(𝑛𝑑 − 1)]
× 100 ] % if 𝑛𝑑 is even
121
122
where
123
nd = the number of nodes in set N,
124
a = the number of arcs in set A,
125
px = the number of predecessors of a node x ,
126
and
sx = the number of successors of node x.
127
128
Some other measures such as total work content (TWK), resource utilization factor and Badiru
129
measure compute schedule complexity as a function of project resource constraints (Nassar et al.
130
2006). We argue that resource constraints form part of the complexity of the project itself and
131
content that schedule quality should be determined independent of such considerations. Since we
6
132
are focusing on the quality of a schedule irrespective of the nature or complexity of project, we
133
exclude these measures from discussion.
134
135
Despite their usefulness the measures mentioned above have several drawbacks. The measure
136
proposed by Nassar et. al (referred to as Cn in this paper) is local in nature i.e., one can only use
137
them to compare two instances of the same project (Nassar et al. 2006). The Davies and
138
Johnson’s measures are variants of the tradition CNCs and count redundant arcs (Latva-Koivisto,
139
2001) thereby giving a false impression of complexity. These measures are focused on the
140
topology of the network only and do not consider the duration of the activities. However, since
141
the duration of the activities is a key determinant for any schedule, any scheduling related
142
analysis using these measures would be incomplete. For example, none of the measures predict
143
the effect of changes in the activity durations on schedule characteristics such as project duration
144
and critical path changes; characteristics that are fundamental for the use of schedules in any
145
system of project controls.
146
147
3.
Centrality Based Quality Measures
148
149
The aim of this paper is to remedy the aforementioned shortcomings of existing complexity
150
based quality measures and develop a meaningful global quality measurement tool for schedules.
151
As outlined earlier, the emphasis of this research is on schedule performance under project
152
changes; hence we define the quality of a schedule as the extent to which the schedule can
153
withstand changes without requiring considerable alterations. We define two new quality
154
measures that rely on the basic schedule information: activity durations and activity precedence.
7
155
A project schedule can be expressed as a weighted directed acyclic graph (DAG). All graphs
156
considered subsequently are AON graphs (with minor modifications our results are equally
157
applicable to activity on the arrow (AOA) networks). A node with in-degree 0 is called a source.
158
A sink is defined as a node with out-degree 0. A classical result in graph theory states that a
159
DAG contains at least one source and one sink (Harary et al. 1965). We can assume that all
160
DAGs considered here have exactly one source and one sink as this a recommended practice for
161
AON schedule networks (multiple sources may be combined using a dummy source and likewise
162
for multiple sinks).
163
Consider a schedule network S (N, A, d) with node set N, arc set A and a duration function d
164
that assigns a duration d(n) to every node (activity) n. The manuscript uses the terms node,
165
vertex, and interchangeably to refer to the constituent tasks of a schedule. Similarly, edge, arc,
166
and precedence are used to reference the dependencies between the tasks. The duration d() of a
167
path  is defined naturally as the sum of durations of nodes that lie on the path. The underlying
168
assumption is that all relationships are of ‘Finish to Start’ type and arcs carry no weights (no
169
lags). Naturally, a node can lie on more than one path in the network. We denote the set of all
170
paths passing through a node n by P(n). Critical path method (CPM) is treated as the standard
171
scheduling technique for construction and related domains. A critical path through a schedule
172
network is typically defined as the longest path connecting the source and the sink. A node
173
(activity) that lies on a critical path is called a critical node (activity). Scheduling for construction
174
projects typically focuses on minimizing the make span to determine the overall duration and to
175
identify early and late start, finish dates of activities for the make span. We define the overall
176
schedule duration as D which is the duration of the longest path in the network.
177
8
178
The use of schedules generated using CPM primarily focuses on management of one or more
179
of the critical paths identified for a network. We argue that a schedule is of good quality if it can
180
accommodate changes in activity durations without resulting in substantial revisions. Thus a
181
good quality schedule should have an overall schedule duration that is stable and is not seriously
182
affected by small changes in the duration of an activity. Further, the critical path should also be
183
stable and a small increase in the duration of an activity should not produce new critical paths.
184
185
We first define the node rank R(n) of each node n as a measure of the relative importance of
186
the node in the schedule network. The node rank determines how close a node is to being critical.
187
𝑁𝑜𝑑𝑒 𝑅𝑎𝑛𝑘 = 𝑅(𝑛) =
188
𝑚𝑎𝑥𝜌∈𝑃(𝑛) 𝑑(𝜌)
𝐷
189
190
191
A higher value of node rank R(n) implies that the node n lies on a relatively longer path
192
joining the source to the sink. A node (activity) with higher value of R(n) is more important as an
193
increase in the duration d(n) is more likely to increase the schedule critical value. Critical nodes
194
have the highest rank of 1 as any change in their duration will increase the critical value. We
195
propose a measure, named Value Centrality Cv, as the average node rank of all nodes in the
196
schedule network.
197
𝑉𝑎𝑙𝑢𝑒 𝐶𝑒𝑛𝑡𝑟𝑎𝑙𝑖𝑡𝑦 = 𝐶𝑣 =
1
∑ 𝑅(𝑛)
|𝑁|
𝑛∈𝑁
198
Here |N| denotes the number of nodes in set N. We can interpret Cv as the average tendency of
199
change in the schedule duration when the duration of an activity is changed. A larger value of Cv
9
200
means that the schedule has lower quality as the schedule duration is very sensitive to changes in
201
activity durations.
202
Our second measure, the path centrality, on the other hand addresses the generation of new
203
critical paths in a schedule network when the duration of an activity is increased. Consider the
204
quantity
∆𝑑(𝑛) = ∆𝑑(𝜌) = 𝐷 − 𝑚𝑎𝑥𝜌∈𝑃(𝑛) 𝑑(𝜌)
205
206
Note that d essentially represents the path float for the longest path through node n. New
207
critical paths cannot be generated by increasing the duration of an already critical node (activity).
208
If a node is not critical then adding d or more to its duration will result in the creation of a new
209
critical path. If we choose a node n randomly and increase its duration by d then a new critical
210
path will be created only if the selected node is non-critical. So if the duration of a randomly
211
selected activity is increased by the amount d, the probability of generating a new critical path
212
is given by
213
𝑃𝑎𝑡ℎ 𝐶𝑒𝑛𝑡𝑟𝑎𝑙𝑖𝑡𝑦 = 𝐶𝑝 =
|𝑛: 𝑛 ∈ 𝑁 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦|
|𝑁|
214
215
Thus a large value of Cp implies that the schedule has low quality as there is a high likelihood
216
that new critical paths will be created if the duration of an arbitrarily chosen activity is increased.
217
These definitions are inspired by social network analysis where various centrality concepts
218
are used to identify the most important actors in the network (Batagelj 1996; Brandes 2001). We
219
interpret our quality measures in a similar manner. If a schedule has many critical tasks then Cp
220
will be small however having a large number of critical nodes will increase Cv. On the other
221
hand having few critical activities would decrease Cv but increase Cp. Furthermore Cv also
222
depends on the duration and location of non-critical activities in the network. If many non10
223
critical nodes lie on nearly critical paths then Cv would be large. Thus intuitively a high quality
224
schedule is one that achieves the best trade-off between the two centrality measures, that is, a
225
schedule for which both Cv and Cp are close to each other and neither is too high.
226
227
3.1
Computational Complexity
228
For a schedule network expressed as a directed acyclic graph, the computational complexity of
229
the critical path method is linear to the number of precedence relationships (or O( |A|) for a
230
graph G(N,A) where N is the set of nodes and A represents the set of arcs) (Alexander et al.
231
1994). The measures of Nassar, and Davies reviewed in this paper can be computed in constant
232
time – O(1). Similarly the path centrality (Cp) and Jhonson’s measure can be computed in linear
233
time and have a complexity of O(N). However, Cv is more complex and has considerable
234
associated computational costs. The computation of Cv has a runtime similar to the CPM
235
algorithm, and unlike the linear CPM algorithm, Cv requires tracking and maintenance of
236
multiple paths that go through each node (exponential complexity). A simplification can be
237
achieved for calculating the Cv without tracking the paths as follows. Since the float for a
238
particular path represents flexibility of the path in comparison to the critical path, using the
239
definition of the float, the node rank can be reformulated as:
𝑅(𝑛) = 1 −
240
241
242
∆𝑑(𝑛)
𝐷
Cv can then be simplified to:
𝐶𝑣 =
1
1
∆𝑑(𝑛)
∑ 𝑅(𝑛) = 1 −
∑
|𝑁|
|𝑁|
𝐷
𝑛∈𝑁
𝑛∈𝑁
243
For G(N,A), the above represents a linear computation of O(A) complexity. However, this
244
simplification comes at the cost of not knowing the possible paths through each node;
11
245
information that has diagnostic and remedial value which is a subject for future research in this
246
area. In addition to the computational benefits, the simplification enables realistic and simplified
247
reasoning.
248
249
3.2
Boundary Conditions and Values
250
Figure 1 shows three theoretical schedules to illustrate the boundary conditions for Cp and Cv.
251
Table 1 presents the measures introduced in this paper – Cp and Cv along with Johonson’s Cj,
252
Davies’ Cd, and Nassar’s Cn measures calculated for the three example schedules. In case of
253
schedule (a), although it is difficult to interpret the quality using Cj and Cd, the measure Cn
254
shows it to be a good schedule as per the ranges specified by Nassar (Nassar et al. 2006). The
255
float for all activities in schedule (a) is 0 and hence Cv =1 which indicates that the slightest
256
change in duration of any activity in this schedule will result in an increased overall duration. On
257
the other hand, Cp for schedule (a) is zero since all activities are critical and no new critical paths
258
can be generated in this network. The combination of Cp and Cv indicates poor quality with a
259
stable critical path for schedule (a) and unreliable schedule duration. It must be noted that
260
schedule (a) is a theoretical case and unlikely to exist in reality since construction projects with
261
any reasonable level of complex interactions between stakeholders require significant parallel
262
execution of tasks.
263
12
S
2
A
2
B
C
3
2
D
F
(a)
A
2
B
2
S
F
265
Figure 1:
2
B
2
S
F
C
15
C
3
D
2
D
2
(b)
264
A
(c)
Activity on Node schedules with each node showing the activity ID and
266
duration: (a) A series schedule with only one path, (b) A schedule with
267
multiple paths with one path being an outright critical path, (c) A schedule
268
with multiple near critical paths
269
270
Schedules (b) and (c) represent more realistic simplifications of construction projects as most
271
schedules can be abstracted to a series of parallel paths (with some shared portions) that have a
272
varying degree of flexibility for each path. However, the dichotomous set of paths shown in (b)
273
and (c) is extreme and unrealistic, and is used here to illustrate the physical meaning of Cp and
274
Cv. Of the three, schedule (b) is a preferred schedule since it represents both the stability of the
275
critical path and a higher redundancy to duration changes of activities (lower Cv). The other
276
measures or Cp alone is not powerful enough to distinguishing between the schedules (b) and (c).
277
Schedule a
Johnson’s
Coefficient
Cj
0
Davies
Coefficient
Cd
0
Nassar’s
Measure
Cn
0
Path
Centrality
Cp
0
Value
Centrality
Cv
1
Schedule b
4
0.3
80%
0.75
0.35
Schedule c
4
0.3
80%
0.75
0.75
Schedule
Name
278
279
Table 1:
Quality measures for schedules in Figure 1
13
280
Although schedule (a) represents a theoretical case where Cv = 1 and Cp = 0, the reverse for
281
these measures is not possible for project schedules modeled as directed acyclic graphs due to
282
the following:
283
284
285
Lemma 1: The value centrality Cv can never be 0 for a network based project schedule
286
Proof: The Cv of a schedule can be zero if and only if all the node ranks are zero. This in turn
287
implies that all activities have zero duration which is impossible.
288
289
We can nevertheless find a practical lower bound for Cv. The value centrality Cv approaches
290
lower values when the float for a large number of tasks approaches the schedule duration. Lower
291
values indicate scenarios where a single path is dominantly longer than the other much smaller
292
parallel paths in the network. By definition of a directed acyclic graph, Cv can never truly be
293
zero since there will always be at least three critical activities with zero float ensuring a tight
294
lower bound of 1⁄|𝑁| for Cv.
295
296
Lemma 2: The path centrality Cp of a network based project schedule can never be equal to 1.
297
Proof: Cp can only be 1 when there are no critical activities in the schedule which is impractical.
298
299
We can readily find a tight upper bound for Cp. A practical schedule network will have at least
300
three critical activities. Thus for a schedule with |N| activities, the tight upper bound on Cp is 1 −
301
1⁄ .
|𝑁|
302
14
303
Based on the above, it can be concluded that the values of Cv lie in the range from 1⁄|𝑁| to 1
304
while Cp is a function with values ranging from 0 to 1 − 1⁄|𝑁|. It is important to note that
305
although these bounds are dependent on |N|, Cp and Cv can be used to compare two schedules S1
306
and S2 with different numbers of activities |N1| ≠ |N2|. Assuming |N2| >> |N1| and that S1 and S2
307
have multiple paths, the lower bound on Cv simply implies that S2 has a range for Cv that is not
308
available for S1 and a clear long path (Cv approaching is lower bound) would imply higher
309
confidence in the quality of S2 (and would be more difficult to achieve for larger schedules).
310
However, the increased range of Cp for S2 implies potential disastrous schedule logic choices
311
that potentially may not plague S1. Near identical Cp and Cv would imply a similar confidence in
312
the stability of the critical path and the schedule duration of the two schedules.
313
314
4.
Software Development and Case Example
315
A software tool was developed as a shared Add-in for MS Project ® to facilitate the computation
316
of the presented measures for project schedules. Algorithms were also implemented for the
317
Davies, Jhonson’s and Nasar’s measures for the sake of easy comparison between the computed
318
coefficients. Figure 2 shows a screen shot of the add-in and the Figure 3 shows the actual use of
319
the add-in for a sample project.
320
15
321
322
Figure 2:
Schedule Quality Measures Add-in showing Cp, Cv, Nassar’s Rank, Davies,
and Johnson’s Coefficients for a Schedule
323
324
325
326
Figure 3:
MS Project Screen-Shot showing the Quality Measures Add-in Integrated
into the Software
16
327
328
A validation case study was carried out using the schedule example (Figure 4) presented in
329
Nassar et al. (2006) as an example of an acceptable schedule. Durations were assumed and CPM
330
calculations were carried out as shown in Table 2 (Cp = 0.4, Cv = 0.98). The addition of duration
331
information reveals information that is missing from the Nassar measure. The combination of the
332
schedule logic and the durations indicates a high likelihood for change in duration of the
333
schedule with a moderate likelihood of creation of new critical paths.
334
335
336
Figure 4:
Simple Schedule for Illustrating the use of Cp/Cv (Nassar et al. 2006)
337
338
17
339
Activity ID
A
B
C
D
E
F
G
H
I
J
Duration ES EF LS LF TF
5
0
5
0
5
0
3
5
8
6
9
1
3
5
8
5
8
0
4
8 12
8 12
0
6
8 14
9 15
1
3
12 15 12 15
0
6
8 14 10 16
2
8
15 23 15 23
0
7
14 21 16 23
2
5
23 28 23 28
0
340
341
342
Table 2:
Assumed durations and CPM calculations for schedule in Figure 4. The
Critical Path is A-C-D-F-H-J. Cp = 0.4 and Cv = 0.98
343
344
A number of simulation runs were carried out to investigate the reliability of the measures. The
345
schedule was simulated using Palisade @Risk ® Software with two possible scenarios. For the
346
first scenario, the duration of all activities were sampled from a triangular distribution with a
347
variation of +/- 2 days from the values tabulated in Table 2. For the second scenario, the
348
durations of the critical activities were fixed and only the durations of the non-critical activities
349
were sampled from the aforementioned distributions. The simulation was designed to keep track
350
of the changes in the critical path as well as any changes in the duration of the schedule. The
351
results of the simulation are shown in Figure 5.
352
18
353
354
(a)
355
356
(b)
(e)
357
358
(c)
(f)
359
360
361
362
363
364
(d)
Figure 5:
Scenario 1: (a) Project Duration, (b) Changes in Critical Path (c) Changes in
Duration. Scenario 2: (d) Project Duration, (e) Changes in Critical Path, (f) Changes in
Duration.
365
366
19
367
5.
368
The measures presented in this paper are not meant to provide tools for generating robust and
369
flexible schedules. The solution to such problems requires considerable domain knowledge,
370
human input and lies beyond the scope of this research. The intent is to enable a consumer of a
371
schedule, given the basic schedule information, to identify logic related problems with any given
372
schedule. The boundary conditions and values help provide a real meaning to the presented
373
measures where path centrality indicates stability of the identified critical path and the value
374
centrality represents the confidence in the computed schedule duration. Although the upper and
375
lower bounds for the presented measures are a function of the number of activities in a particular
376
network, the measures are truly global in nature and allow for comparison between different
377
schedules of varying sizes for different projects. As presented, the combination of Cp and Cv can
378
distinguish between schedules that cannot be discriminated based on the existing complexity
379
measures that focus on network topology alone.
380
Limitations of Path and Value Centrality
Although the measures presented in this paper provide an assessment of the overall
381
quality of schedules, remedial actions cannot be taken based on the information presented by
382
these measures. Future research in this area can provide diagnostic capabilities by identifying
383
specific problem areas within a schedule by developing aggregate measures for path segments in
384
a schedule; information that can aid the domain experts in the complex decision making required
385
for remedial actions.
386
387
6.
Conclusions
388
The measures addressed in this research provide a tool for a stakeholder, equipped with the basic
389
schedule information, to make a judgment about the “quality” of a schedule. The measures are
20
390
not intended as a test of the scheduler’s ability; rather, the measures are intended to enable
391
identification of schedules (and problem portions of such schedules) that would consistently
392
require major changes in response to even the slightest change in the activity durations or logic.
393
The scheduler or a domain expert, equipped with additional knowledge, can then make the
394
necessary alterations to the schedule and rectify the problem areas. An MS Project based add-in
395
has been developed to facilitate the use of these measures on real project schedules. Future work
396
of the authors is exploring the use of Eigen Value centrality measures for predicting schedule
397
behavior under change.
398
399
Acknowledgements
400
The authors thank King Fahd University of Petroleum and Minerals (KFUPM), Dhahran, Saudi
401
Arabia for its continuous support of their research. This research was funded by the Deanship of
402
Scientific Research at KFUPM under Research Grant IN101021.
403
References
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
Alexander, C., Reese, D., and Harden, J. (1994). "Near-critical path analysis of program activity
graphs." Proc., Proceedings of the 2nd International Workshop on Modeling, Analysis,
and Simulation of Computer and Telecommunication Systems, January 31, 1994 February 2, 1994, Publ by IEEE, 308-317.
Badiru, A. B., and Pulat, P. S. (1995). Comprehensive project management : integrating
optimization models, management principles, and computers, Prentice Hall PTR,
Englewood Cliffs, N.J.
Batagelj, V. (1996). "Centrality in Social Networks." Developments in Data Analysis, A.
Ferligoj, and A. Kramberger, eds.Ljubljana, Slovenia, 100-109.
Brandes, U. (2001). "A faster algorithm for betweenness centrality." Journal of Mathematical
Sociology, 25(2), 163-177.
Davies, E. M. (1977). An experimental investigation of resource allocation in multiactivity
projects with resource restraints, University of Birmingham, Environmental Modelling
and Survey Unit, Birmingham.
De Reyck, B., and Herroelen, W. (1996). "On the use of the complexity index as a measure of
complexity in activity networks." Eur. J. Oper. Res., 91(2), 347-366.
Dossick, C. S., and Schunk, T. K. (2007). "Subcontractor schedule control method." J. Constr.
Eng. Manage., 133(3), 262-265.
21
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
Freeman, L. C. (1977). "A Set of Measures of Centrality Based on Betweenness." Sociometry,
40(1), 35-41.
Galloway, P. D. (2006). "Survey of the construction industry relative to the use of CPM
scheduling for construction projects." J. Constr. Eng. Manage., 132(7), 697-711.
Harary, F., Norman, R. Z., and Cartwright, D. (1965). Structural models: an introduction to the
theory of directed graphs, Wiley, New York,.
Khan, M. A., and Siddiqui, M. K. (2010). "Quantitative Assessment of Network Based
Schedules." Proc., International Conference on Computing in Civil and Building
Engineering ICCCBE 2010, 273.
Latva-Koivisto, A. M. (2001). "Finding a complexity measure for business process models."
Helsinki University of Technology, Systems Analysis Laboratory, Espoo, Finland.
Nassar, K. M., and Hegab, M. Y. (2006). "Developing a complexity measure for project
schedules." J. Constr. Eng. Manage., 132(6), 554-561.
Palisade (2000). "@Risk for Project Professional." <www.palisade.com/riskproject/>. (March
2012).
Russell, A. D., and Udaipurwala, A. (2000). "Assessing the quality of a construction schedule."
Proc., Construction Research Congress VI, American Society of Civil Engineers, 928937.
Siddiqui, M. K., and O’Brien, W. J. (2009). "A Formal Mapping Based Approach for
Distributed Schedule Coordination on Projects." Adv. Eng. Inf., 23(1), 104-115.
Smith, S. F. (2005). "Is Scheduling a Solved Problem?" Multidisciplinary Scheduling: Theory
and Applications, G. Kendall, E. Burke, S. Petrovic, and M. Gendreau, eds., Springer US,
3-18.
Zack, J. G., Jr. (1992). "Schedule 'games' people play, and some suggested 'remedies'." J.
Manage. Eng., 8(2), 138-152.
22