We study the project budget version of the stochastic discrete time/cost trade-off problem (SDTCTP-B) from the viewpoint of the robustness in the scheduling. Given the project budget and a set of activity execution modes, each with uncertain activity time and cost, the objective of the SDTCTP-B is to minimize the expected project makespan by determining each activity’s mode and starting time. By modeling the activity time and cost using interval numbers, we propose a proactive project scheduling model for the SDTCTP-B based on robust optimization theory. Our model can generate robust baseline schedules that enable a freely adjustable level of robustness. We convert our model into its robust counterpart using a form of the mixed-integer programming model. Extensive experiments are performed on a large number of randomly generated networks to validate our model. Moreover, simulation is used to investigate the trade-off between the advantages and the disadvantages of our robust proactive project scheduling model.
In project management, the project duration can usually be shortened by allocating more resources to critical activities. The number of resources tends to be discrete, such as the numbers of workers and machines. These resources are usually treated as nonrenewable resources and measured by capital (or cost), resulting in the discrete time/cost trade-off problem (DTCTP) [
In the deterministic DTCTP, each activity has multiple execution modes that are characterized by specific time and cost combinations. In terms of the types of the objective function, the DTCTP can be divided into three versions: the deadline problem (DTCTP-D), the budget problem (DTCTP-B), and the time/cost trade-off curve problem (DTCTP-C). In the DTCTP-D, given a set of modes and a project deadline, the objective is to minimize the total project cost by specifying for each activity an execution mode. In the DTCTP-B, a project budget is given and the objective is to determine the modes that minimize the project makespan. In the DTCTP-C, the goal is to determine the Pareto curve that minimizes the project cost and makespan simultaneously.
Once the mode of each activity is determined, we can determine the baseline schedule
However, during project execution, due to considerable uncertainties, the optimal baseline schedule which is obtained based on a deterministic environment and complete information may deviate from our expectations or even become unfeasible. Possible sources of uncertainties may be a shortage of machineries, a delayed delivery of materials, the absence of workers, fluctuations in the exchange rates, and so forth [
Recent studies have paid more attention to the stochastic DTCTP (SDTCTP), which accounts for uncertainties by treating the time and cost of activities as stochastic variables, with the objective of optimizing the expected project performance. As early researchers in the field, Gutjahr et al. [
However, the above-mentioned research papers primarily focused on optimizing the system performance in an average sense and these prior approaches cannot guarantee the performance of the baseline schedule during a single project execution. Therefore, determining a robust baseline schedule under uncertainty is increasingly attracting the attention of scholars. To achieve a robust baseline schedule, the use of robust optimization is a natural choice. Robust optimization can determine a solution with certain robustness by optimizing the worst-case performance of the system. Although robust optimization has been used to solve some classic project scheduling problems [
To the best of our knowledge, addressing both time- and cost-uncertainty and applying robust optimization in solving the SDTCTP-B have not been taken into account in both the project scheduling and the robust optimization literature. The contributions of this paper are as follows. We proposed a proactive scheduling model for the SDTCTP budget problem (SDTCTP-B) based on robust optimization theory. Our model uses interval numbers to model the uncertain time and cost of the activities that can follow any type of probability distribution. The objective of our model is to generate a stable baseline schedule that can account for some of the uncertainties during project execution to ensure, to the extent possible, that each activity begins at their respective planned start time. We conducted a detailed experimental analysis for our proposed model. We used experimental design to randomly generate a large number of instances to validate our model. In addition, robust optimization improves the schedule stability at the cost of prolonging the project duration. Therefore, we used simulation to investigate the trade-off between the advantages and the disadvantages of robust optimization. Specifically, we analyzed the impact of the number of activities, the network order strength, and the number of modes on the schedule stability by using discrete systems simulation.
This paper is organized as follows. Section
The stochastic discrete time/cost trade-off problem can be described as follows. A project network
Given the project budget
Faced with the uncertainty in the duration and cost of each activity, the baseline schedule generated by the above deterministic model is not expected to be executed as determined, thereby resulting in failure to achieve the desired project objective. When the uncertainty is considered, we notice that the uncertain parameters mainly affect constraints (
In practice, it is usually easier for decision-makers to estimate the range and the most likely value of the duration and cost of activities rather than their probability distribution. Therefore, we use interval numbers to model the uncertain duration and cost of the activity. For mode
For each activity
Our proactive scheduling model for SDTCTP-B based on robust optimization [
The above model introduces the uncertain parameters into the deterministic DTCTP model and is able to generate robust solutions.
Our model has two primary advantages. The first advantage is that the robustness level of the obtained schedule can be freely adjusted. The greater the value of the parameter
The constraints of problem (
The dual of problem (
Similarly, the “max” part of (
Then we can obtain the equivalent mixed-integer linear optimization model of Model 1 by substituting (
We use an example project network in Figure
The example project network.
Table
Computations for the example problem.
|
Mode | Total cost | Makespan | Feasible | |
---|---|---|---|---|---|
Activity 3 | Activity 4 | ||||
0 | 1 | 1 | 86 | 7 | No |
1 | 2 | 77 | 9 | No | |
2 | 1 | 74 | 13 | Yes | |
2 | 2 | 65 | 13 | Yes | |
|
|||||
1 | 1 | 1 | 86 | 7 | No |
1 | 1* | 89 | 7 | No | |
1 | 2 | 77 | 9 | No | |
1 | 2* | 80 | 10 | No | |
1* | 1 | 90 | 9 | No | |
1* | 1* | 93 | 9 | No | |
1* | 2 | 81 | 9 | No | |
1* | 2* | 84 | 10 | No | |
2 | 1 | 74 | 13 | Yes | |
2 | 1* | 77 | 13 | No | |
2 | 2 | 65 | 13 | Yes | |
|
|
|
|
|
|
2* | 1 | 78 | 15 | No | |
2* | 1* | 81 | 15 | No | |
2* | 2 | 69 | 15 | Yes | |
2* | 2* | 72 | 15 | Yes | |
|
|||||
2 | 1* | 1* | 93 | 9 | No |
1* | 2* | 84 | 10 | No | |
2* | 1* | 81 | 15 | No | |
2* | 2* | 72 | 15 | Yes |
As shown in Table
We have randomly generated a large number of problem instances to validate our model. Section
Parameter settings.
Parameters | Value |
---|---|
Number of activities |
10; 20; 30 |
Order strength (OS) | 0.3; 0.5; 0.7 |
Number of modes |
4; 8 |
Most likely activity duration | Uniformly drawn from |
Most likely smallest activity cost | Uniformly drawn from |
Activity cost slope | Uniformly drawn from |
|
Uniformly drawn from |
|
0.5 |
Specifying 3 settings for the number of activities, 2 settings for the number of execution modes, and 3 settings for OS, we generated 10 problem instances for each of the 3 × 2 × 3 parameter settings, resulting in 180 instances in total.
We also need to generate the most likely value of duration and cost for each activity. In DTCTP, the types of cost functions can be linear, convex, concave, or random. We study the random case, which is more general. Following Demeulemeester et al. [
Our model deals with uncertain data, so we generated the maximum deviation of activity duration and cost by letting
The parameter
Computational results.
|
OS |
|
Γ(0) | Γ(1) | Γ(2) | Γ(3) | Γ(4) | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Avg | Min | Max | % | Avg | Min | Max | % | Avg | Min | Max | % | Avg | Min | Max | % | Avg | Min | Max | % | |||
10 | 0.3 | 4 | 62.3 | 53 | 76 | 100 | 83.3 | 68 | 101 | 100 | 100.4 | 74 | 116 | 100 | 116.1 | 82 | 154 | 100 | 131 | 96 | 168 | 100 |
8 | 45.4 | 40 | 51 | 100 | 55.6 | 49 | 65 | 100 | 62.4 | 56 | 70 | 100 | 69.9 | 62 | 78 | 100 | 77.3 | 67 | 91 | 100 | ||
0.5 | 4 | 69.3 | 50 | 86 | 100 | 89.9 | 63 | 110 | 100 | 108.8 | 68 | 127 | 100 | 126.2 | 69 | 159 | 100 | 144.2 | 87 | 179 | 100 | |
8 | 58.8 | 51 | 67 | 100 | 75.4 | 69 | 88 | 100 | 86.9 | 80 | 95 | 100 | 95.1 | 83 | 108 | 100 | 102 | 90 | 118 | 100 | ||
0.7 | 4 | 77.6 | 64 | 90 | 100 | 109.5 | 91 | 122 | 100 | 129.7 | 115 | 144 | 100 | 152.6 | 139 | 172 | 100 | 175.1 | 151 | 202 | 100 | |
8 | 76.6 | 66 | 86 | 100 | 94.2 | 82 | 110 | 100 | 106.8 | 96 | 122 | 100 | 117.9 | 102 | 133 | 100 | 127.6 | 109 | 148 | 100 | ||
|
||||||||||||||||||||||
20 | 0.3 | 4 | 73.1 | 60 | 86 | 100 | 85.1 | 69 | 96 | 100 | 95 | 78 | 107 | 100 | 106.1 | 89 | 120 | 100 | 114.3 | 94 | 127 | 100 |
8 | 62.5 | 56 | 68 | 100 | 70.1 | 64 | 75 | 100 | 75.5 | 66 | 81 | 80 | 80.4 | 70 | 86 | 70 | 85 | 71 | 94 | 70 | ||
0.5 | 4 | 92.5 | 65 | 118 | 100 | 112.4 | 76 | 140 | 100 | 125.8 | 84 | 160 | 100 | 139.2 | 93 | 179 | 100 | 153 | 102 | 201 | 90 | |
8 | 83 | 74 | 95 | 100 | 94.4 | 82 | 108 | 90 | 101.2 | 85 | 119 | 80 | 108.4 | 91 | 127 | 70 | 114.9 | 98 | 135 | 80 | ||
0.7 | 4 | 124.5 | 106 | 147 | 100 | 151.3 | 126 | 172 | 100 | 168 | 140 | 191 | 100 | 185.4 | 156 | 220 | 100 | 198.3 | 166 | 234 | 100 | |
8 | 119 | 106 | 137 | 100 | 137.7 | 124 | 154 | 80 | 148.4 | 134 | 164 | 70 | 157.6 | 141 | 173 | 70 | 166.2 | 147 | 182 | 60 | ||
|
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30 | 0.3 | 4 | 76.5 | 71 | 81 | 100 | 87.1 | 79 | 98 | 80 | 95.5 | 87 | 110 | 100 | 103.4 | 95 | 114 | 70 | 108.5 | 98 | 123 | 80 |
8 | 68.1 | 63 | 79 | 100 | 76.4 | 70 | 89 | 40 | 80.8 | 72 | 95 | 10 | 84.5 | 75 | 101 | 20 | 88.8 | 80 | 103 | 0 | ||
0.5 | 4 | 106.2 | 88 | 124 | 100 | 121.5 | 102 | 139 | 60 | 131.6 | 109 | 156 | 30 | 142.3 | 122 | 172 | 0 | 152.4 | 130 | 189 | 20 | |
8 | 91.3 | 83 | 102 | 90 | 102.6 | 93 | 115 | 10 | 107.9 | 98 | 120 | 0 | 113.6 | 101 | 128 | 0 | 119.1 | 106 | 136 | 0 | ||
0.7 | 4 | 153.3 | 124 | 166 | 100 | 177.6 | 141 | 191 | 40 | 192.3 | 150 | 207 | 40 | 207.8 | 159 | 232 | 20 | 220.8 | 173 | 249 | 30 | |
8 | 128.9 | 113 | 137 | 80 | 145.7 | 129 | 158 | 0 | 154.5 | 139 | 172 | 20 | 163.2 | 150 | 189 | 0 | 171.4 | 159 | 204 | 0 |
The results in Table
Small problem instances (
Robust optimization improves the schedule robustness at the expense of prolonging the project duration. Therefore, we are interested in the trade-off between the advantages (improved schedule stability) and the disadvantages (increased project duration) of robust optimization. Specifically, from the viewpoint of stability cost, we use simulation to investigate the impact of project network structure parameters (i.e., the number of activities, the order strength, and the number of modes) on the schedule stability.
We use the
It is not realistic to use a full factorial experiment to analyze the impact of different factors on the schedule stability [
The simulation strategy is as follows. (1) The weights
For each instance in our data set, the number of simulation replications is set to 100. Note that the stability costs mentioned in the following are always the average value.
Figure
Impact of the number of activities on the schedule stability (
For each robustness level
We see that although the actual difference between
Figure
Impact of the order strength on the schedule stability (
An interesting finding is that when
Figure
Impact of the number of modes on the schedule stability (
Moreover, in most cases, for any factor (the number of activities, the order strength, or the number of modes), Figures
In this paper, we presented a proactive scheduling model for the project budget version of the stochastic discrete time/cost trade-off problem based on robust optimization theory. Computational experience on the randomly generated problem dataset validated our model. We also used simulation to analyze the impact of different project network parameters on the schedule stability.
From the experiments in the stochastic time/cost trade-off environment, we conclude that (1) a larger value of the robustness level parameter
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the reviewers for providing valuable suggestions that have improved the quality of this paper. This research was supported by the National Science Foundation of China under Grant 71271019, the Humanities and Social Sciences Foundation of the Ministry of Education under Grant 12YJA630158, and the Soft Science Research Program of Shanghai under Grant 14692105900.