Artif Intell Rev (2009) 32:115–123 DOI 10.1007/s10462-009-9138-1
Effects of project size and resource constraints on project duration through priority rule-base heuristics Recep Kanit · Omer Ozkan · Murat Gunduz
Published online: 22 October 2009 © Springer Science+Business Media B.V. 2009
Abstract Priority rules are one of the frequently used methods in project programming with resource-constraints. In this paper, the effects of project size and number of resource constraints on project duration are compared to the performances of pre-selected priority rules. Ten projects in different sizes have been programmed with 3, 5, 7, 9, and 11 limitedresource conditions by means of MRPL (Maximum Remaining Path Length), LFT (Latest Finish Time), MNSLCK (Minimum Slack Time), EFT (Earliest Finish Time), and LST (Latest Start Time) priority rules. When the number of resource constraints is low, the performance of MRPL is generally observed to be higher. As the number of resource constraints increases, a decrease in the performance of MRPL is observed in contrast with an increase in the performance of LFT. Keywords
Heuristic · Priority rules · Project programming · Resource constraints
1 Introduction The Critical Path Method (CPM) is a method that has been widely used by practitioners for planning and controlling large-scale projects in manufacturing and construction industries. A traditional CPM analysis assumes unlimited resources. However, during the scheduling process of a real project, it is a necessity to use limited resources such as limited crew sizes,
R. Kanit Department of Technical Education, Gazi University, Ankara, Turkey e-mail:
[email protected] O. Ozkan Construction Department, Sakarya University, 54187 Sakarya, Turkey e-mail:
[email protected] M. Gunduz (B) Department of Civil Engineering, Middle East Technical University, Ankara 06531, Turkey e-mail:
[email protected]
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limited equipment amounts, and limited materials (Leu et al. 1999). Resource-constrained scheduling problems have been studied intensively in the construction and manufacturing industries as a result of practical applications; and in the light of the findings of these studies, several heuristic models have been generated to solve resource-constrained scheduling problems. A primary aim of this paper is to quantify the effects of pre-selected heuristic models on the project duration estimates. In order to achieve this aim, ten projects in different sizes have been programmed with various limited-resource conditions using pre-selected priority rules which have been known to demonstrate high performance in the literature. Furthermore, it is assumed that comparing the performances of these priority rules could generate additional management information, such as the impact of resource constraints on project scheduling, the sensitivity level of project size and the combinative effects. In short, this study is expected to give project management an insight into the project schedule.
2 Literature review Resource-constrained scheduling models can be divided into two categories as deterministic scheduling models and nondeterministic scheduling models. Currently, resource-constrained scheduling models mostly focus on deterministic situations. It is worth mentioning here that analytical and heuristic methods are the most popular techniques of deterministic resourceconstrained scheduling. Early attempts to solve deterministic resource-constrained scheduling problems used mathematical models such as Integer linear programming, dynamic programming, as well as branch and bound in order to find an optimal solution (David 1973). For instance, Elmaghraby (1977) and Talbot (1982) used integer programming techniques to solve a resource-constrained problem under a certain environment. A great deal of computational effort is required to solve problems of this kind. Therefore, to avoid the problem of “combinatorial explosion” heuristic rules were also used to solve such problems (Morse and Whitehouse 1988; Tsai and Chiu 1996). To date, numerous heuristic scheduling rules have been proposed to solve deterministic resource-constrained scheduling problems, such as the MINSLCK model (David and Patterson 1975), the ROT model (Elsayed 1982), and the GENRES model by Whitehouse and Brown (1979). Branch and Bound compared the performances of certain projects with 20, 30, 50, 100, and 250 activities using heuristic algorithms. Sung and Lim (1997) observed an increase in speed and performance as a result of an increase in the number of project activities. Further research on the issue has revealed that the effectiveness of the priority rules used in heuristic methods highly depends on the project size and the number of available resources. In a study, Abbasi and Arabiat (2000) tested 60 projects in which the cash flow was defined by a priority rule that can be described as a combination of the LST and SPT methods. This new method, namely LSSPT (late start and minimum activity time), provided a higher performance (Abbasi and Arabiat 2000) when compared to some other priority rules such as latest start time (LST), or latest finish time (LFT). David and Patterson (1975) examined the success ratio of various priority rules in 83 different projects. According to the findings of this study, the MNSLCK and LFT priorities gave the best result in 24 and 12 projects, respectively. In their study which used non-renewable resources, Ozdamar and Ulusoy (1996) made a similar analysis on 78 projects through parallel methods according to four different priority rules. MNSLCK gave the best result in 24 projects and LFT in 22 projects.
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Table 1 Priority rules in literature Priority rule
Order
Priority rule
Maximum remaining path length (MRPL) Latest start time (LST) Early start time (EST) Early finish time (EFT) Latest finish time (LFT) Shortest processing time (SPT) Most total successors (MTS) Greatest rank positional weight (GRPW) Late start and shortest processing time (LSSPT) Minimum slack time (MNSLCK)
Max Min Min Min Min Min Max Max Max Min
T pE − TiE − Si j LS j ES j EF j LF j dj Fj d j + i∈ F j d j Fi j = T + di j LS j − ES j
d j (activity time), F j (activity finish time), T pE (project time), TiE (activity start time) Si˙j (Slack)
Schirmer (1999) scheduled his projects with 30 activities and used eight priority rules. Out of these eight priority rules, the LFT priority yielded the highest level of performance in 42 projects whereas the MNSLCK priority gave the best result in 24 projects and the MTS priority in 23 projects. Klein (2000) used various priority rules for his study and found out that the LFT priority gave the best performance in 36 projects, the MTS Priority in 20 projects, and the GRPW and LST priorities in 19 projects. When the parallel methods were used, the priority which gave the best performance was the LFT priority with 28 projects. Each heuristic model has its own philosophy, but they all try to increase the possibility of obtaining the best solution. The most commonly used priority rules in the literature and their definitions are given in Table 1. In the first column of the table, the name of each rule is given. The second column denotes whether the rule is applied to a particular job with the smallest ( min) or the largest (max) priority value. Finally, the last column describes the computation of the priority rule.
3 Project algorithm and priority rules The priority rules used in resource-constrained conditions give various results depending on the size of a project, the number of resources and the quantity of limitation. Keeping this in mind, 10 projects in different sizes have been programmed with various limited-resource conditions in this study using pre-selected priority rules which have been known to demonstrate high performance in the literature. These pre-selected priority rules are MRPL (Maximum Remaining Path Length), LFT (Latest Finish Time), MNSLCK (Minimum Slack Time), EFT (Earliest Finish Time), and LST (Latest Start Time). The priority rules used in this study are explained below: (a) MRPL is an abbreviation for “Maximum Remaining Path Length”. The time between the start and end of a process is taken into consideration and calculated for each process with the Eq. 1 below: MRPL = T pE − Ti E − Si j
(1)
where; T pE : Projects time; Ti E : Activity start time; Si j : Slack
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The calculated MRPL time leads to programming the greater process. Thus, the processes are programmed according to this priority in order. It is worth mentioning here that this priority rule was first used by Brooks (Brooks and White 1965). (b) LFT is the abbreviation of “Latest Finish Time”. The latest finish time for each project is calculated, and the process with the smallest value of LFT is programmed first. This priority rule was first used by Ahuja (1976). (c) MNSLCK is the ampleness amount of processes. This term can also be defined as the importance level of processes. The ampleness of each process is calculated (shown in Eq. 2), and the process with the least ampleness is programmed first. David and Patterson (1975) used this priority rule in their studies to find the optimum project time. Si j = LSi j − ESi j
(2)
(d) EFT is the abbreviation of “Earliest Finish Time”. The earliest finish time for each project is calculated, and the process with the smallest value of EFT is programmed first. (e) LST is the abbreviation of “Latest Start Time”. The latest start time for each project is calculated, and the process with the smallest value of LST is programmed first. The program developed by the authors used 3, 5, 7, 9, and 11 resource-constraints one at a time to make an investigation into the effects of the number of resource-constraints on project duration. For each project, the priority rule that minimized the project duration was accepted to be successful. The commonly known priority rules with implementation histories are used in the algorithm shown in Fig. 1.
4 Project programming Project programming consists of two phases. In the first phase, the project’s CPM solution is programmed without any kind of source limitation. Then, for each activity, early start (ES), early finish (EF), lately start (LS), lately finish (LF), start (S), and finish time (F) of different processes are calculated. In the second phase, the priorities (MRPL, LFT, MNSLCK, EFT, and LST) that will be used by processes are calculated. The processes are consequently programmed in order, according to these priorities. The program also takes into account the daily resource requirements and therefore checks each activity’s resource requirement on a daily basis. In order to get better results about the effects of the number of resource-constraints on project duration, 11 different resource-constraints were defined as 3, 2, 4, 5, 2, 2, 5, 3, 4, 4, 2 units/day, in a sorted order, respectively. An analysis was run after dividing the resource constraints into five groups. For the first group, only the first three resource-constraints were used whereas for the second group five, for the third group seven, for the fourth group nine, and finally for the fifth group 11 resource constrains were used. During the processing stage of the projects, if the required resource was more than the available one, the activity was delayed to the following day. The process continued until the finish time for the last activity was calculated. After all these steps, the priority rule with the least project duration was accepted to be the most powerful one. The results of the limitless resource and resourceconstrained conditions of the projects are given in Table 2. The values typed in bold are the minimum project duration values. It can be concluded from this table that there were more than one successful method in some cases. A detailed interpretation of the results will be given in the next section.
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Read Data
Select the priority rule to be used
Reset time to zero; maximize the available resource amount
Construct the activity list
Choose the unscheduled activity with the highest priority
Y Is the activity research demand enough?
N Calculate the amount of remaining resources
Schudule the activity
Calculate the start and finish times
Activitiy finish time (t)
N Calculate the amount of remaining resources
Y Last activity or not?
Scheduled each activity?
N Y Schedule the following activity which has the priority
Terminate
Fig. 1 Priority rule algorithm
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Table 2 Project programming results Project number
Activity number
CPM Duration (day)
1
10
31
2
3
4
5
6
7
8
123
11
12
17
19
23
25
31
33
31
39
43
50
59
65
Number of constraints
MRPL (day)
LFT (day)
MNSLCK (day)
EFT (day)
EST (day)
3
42
42
42
41
42
5
55
53
59
53
54
7
61
63
69
65
63
9
65
64
66
66
66
11
67
66
69
69
68
3
45
54
54
47
49
5
63
63
72
63
64
7
72
71
75
73
73
9
73
73
73
78
81
11
77
79
77
80
81
3
48
48
48
48
51
5
59
59
57
55
57
7
65
65
62
67
62 70
9
68
68
68
69
11
73
71
71
75
77
3
58
53
55
59
53
5
65
65
71
67
67
7
71
74
76
77
76
9
88
94
88
89
92
11
92
92
96
96
97
3
65
71
76
69
73
5
84
92
92
90
89
7
89
94
94
90
89
9
91
98
97
99
98
11
95
97
97
95
101
3
87
96
91
87
90
5
96
101
103
100
105
7
101
101
103
101
105
9
111
108
107
109
107
11
121
119
116
118
116
3
100
101
105
100
99
5
113
119
115
117
115
7
117
119
117
118
117
9
124
122
125
125
127
11
128
122
125
126
124
3
115
116
120
117
117
5
131
130
141
133
135
7
135
133
141
135
138
9
141
140
142
138
138
11
145
140
144
140
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Effects of project size and resource constraints
121
Table 2 continued Project number
Activity number
CPM Duration (day)
9
36
70
10
37
Number of constraints
74
MRPL (day)
LFT (day)
MNSLCK (day)
EFT (day)
EST (day)
3
123
124
131
125
127
5
147
154
161
147
149
7
151
152
157
150
153
9
159
159
163
160
161
11
165
165
167
169
167
3
136
139
146
137
141
5
161
165
161
163
163
7
164
164
166
163
167
9
177
174
177
174
181
11
185
179
185
182
181
5 Research outcomes The results of all scheduling combinations are presented in Table 2 in which the projects with the minimum durations are given in bold. A thorough reading of the table shows that for some cases there is more than one successful rule. Table 3 presents the number of successful rules assuming that a rule is successful when there are a number of minimum project durations. It can also be deduced from this table that when the number of resource-constraints is low, MRPL outperforms other rules. On the other hand, when the number of constraints is increased, the performance of MRPL reduces significantly. On the contrary, LFT performance increases when the number of resource-constraints is increased.
Table 3 Success ratios of priority rules Number of constraints
MRPL
LFT
MNSLCK
EFT
EST
3
8
2
1
4
1
5
6
3
1
4
0
7
6
4
2
3
3
9
5
6
3
2
2
11
4
7
3
2
2
Assuming each rule is successful when minimum project duration is same Table 4 Success ratios of priority rules Number of constraints
MRPL
LFT
MNSLCK
EFT
EST
3 5 7 9 11
5 3 2 1 0
0 1 2 2 3
0 0 0 0 0
1 2 1 0 0
0 0 0 0 0
Assuming no rule is successful when minimum project duration is same
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Standardized Duration
122 2.5 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1
3 Constraints 5 Constraints 7 Constraints 9 Constraints 11 Constraints
5
10
15
20
25
30
35
40
Number of Activities Fig. 2 Number of activities vs. standardized duration
Unlike Tables 3, 4 represents the number of successful rules giving minimum project duration alone. MRPL almost always gives the minimum project duration when the number of resource-constraints is three. However LFT performance is observed to be better than the other rules when the number of resources is increased. 5.1 Effect of number of activities on project duration First of all, each minimum project duration for each resource constraint is divided by the limitless resource CPM duration in order to standardize the duration (Eq. 3). Standardized duration = Minimum project duration /Original CPM duration
(3)
In order to see the effects of the number of activities on project duration, the standardized duration is compared with the number of activities in Fig. 2. As can be concluded from this figure, there is no clear indication that project duration increases in direct correlation with the number of activities and resource-constraints. For instance, when the number of constraints is three, the increasing trend can be seen; but when the number of constraints is higher, the increasing trend is lost. Furthermore, the standardized durations are close to each other and follow a similar trend when the number of activities is 20 and higher. It can also be seen that the standardized durations almost follow a constant trend for nine and 11 constraints. The difference between the highest number of activities and the lowest for seven, nine, and 11 constraints are low, whereas this value gets bigger for three and five constraints. There has been a slight increase in MNSLCK and a minor decrease in EFT performance. The performance of EST is observed to be random.
6 Conclusion Priority rules are commonly used in programming the projects where there are high numbers of activities and resources. This study investigated the effects of the project size and number of resources on the performances of certain pre-selected priority rules. After running
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250 different scheduling schemes with various numbers of constraints and activities, it was seen that there exists a significant correlation between the number of constraints and the pre-selected priority rules. Moreover, MRPL performance was observed to be higher for low numbers of constraints, whereas LFT performance was significantly higher when the number of constraints increased. Finally, there is a slight increase in project duration when the number of activities is increased for small numbers of constraints. However, there is no clear trend of increase in project duration, when number of constraints is high.
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