Priority rule method for random flexible project scheduling problem

Abstract

The invention relates to the technical field of project scheduling optimization, in particular to a random flexible project scheduling problem-oriented priority rule method, which comprises the following steps of: for a resource-limited project scheduling problem, performing priority calculation by using activity construction period information when performing selection and sorting decision on residual activities by using a priority rule; solving a random flexible project scheduling problem in the resource-constrained project scheduling problem by adopting a method based on a priority rule; in order to compare and analyze the performance of the priority rule in solving the random flexible resource constrained project scheduling problem, an example covering different scheduling environments is constructed based on a standard test example, and an experimental result is obtained; according to the priority rule method for the random flexible project scheduling problem, the performance of the method in solving the flexible RCPSP with the random construction period and reworking is tested through a classical priority rule for activity selection and activity sorting, and more comprehensive reference is provided for engineering actual project scheduling.

Description

Technical Field

[0001] The present invention relates to the technical field of project scheduling optimization, and in particular to a priority rule method for random flexible project scheduling problems. Background Art

[0002] Traditional resource-constrained project scheduling research typically assumes a deterministic and acyclic project network structure. However, in engineering practice, some activities within a project have conflicting and interdependent relationships, creating a flexible project network. For example, in the manufacturing of customized equipment, some parts can be fabricated in-house or outsourced. If outsourcing is chosen, the decision must be made whether to ship the parts directly to the customer site or have them assembled at the factory. If fabricated in-house, subsequent activities include quality inspection, packaging, and shipment to the customer site. If the parts are outsourced and shipped directly to the customer site, an outsourcing acceptance check is required before assembly and commissioning. Clearly, scheduling decisions for projects with flexible project networks must consider not only activity sequencing but also activity selection.

[0003] In addition to the characteristics of flexible project networks, actual engineering project scheduling also needs to consider the numerous uncertainties encountered during project execution. For example, new product design and R&D projects face uncertainties such as new technology development, unclear customer needs, urgent orders, and equipment failures. Therefore, this paper, while considering flexible project networks, further incorporates random activity durations and random rework. The traditional resource-constrained project scheduling problem (RCPSP) has been proven to be strongly NP-hard.

[0004] For RCPSP with flexible project networks, German scholars Kellenbrin and Helber first studied this type of problem with the engineering background of aircraft engine maintenance, constructed a mathematical model and proposed a genetic algorithm based on dual chromosomes.

[0005] Regarding RCPSPs that simultaneously consider stochastic factors and flexible project networks, only a few studies have addressed this issue. A mathematical model embedded with random chance constraints was constructed for this problem, and a hybrid heuristic algorithm based on the sample average approximation and an artificial algae algorithm based on population evolution was developed. A corresponding constraint programming model was constructed, maximizing time and resource balance as the optimization objectives for this problem. Currently, no research has considered both stochastic rework and flexible project networks. For RCPSPs with stochastic duration and rework, a mathematical model based on a continuous-time Markov decision process was developed, and the optimal strategy for solving the problem was determined using stochastic dynamic programming. Summary of the Invention

[0006] The purpose of this paper is to propose a priority rule method for the random flexible project scheduling problem. By using classic priority rules for activity selection and activity sorting, its performance in solving flexible RCPSP with random duration and rework is tested, providing a more comprehensive reference for actual engineering project scheduling.

[0007] To achieve this object, the present invention adopts the following technical solutions:

[0008] A priority rule method for stochastic flexible project scheduling problem includes the following steps:

[0009] For resource-constrained project scheduling problems, projects are usually represented by a single-code network graph G(N,A), where N = {1, 2, ..., n} is the set of activities in the project and A represents the precedence constraint relationship between activities.

[0010] When using the priority rule to select and sort the remaining activities, the activity duration information is used in two ways: for activities without rework, the average duration is directly used for priority calculation; for activities with rework, the remaining duration of the activities with rework is estimated by aggregating the remaining duration and then the priority is calculated;

[0011] A method based on priority rules is used to solve the stochastic flexible project scheduling problem in resource-constrained project scheduling. The priority rules include activity selection priority rules and activity sorting priority rules.

[0012] In order to compare and analyze the performance of the aforementioned priority rules in solving the stochastic flexible resource-constrained project scheduling problem, a case covering different scheduling environments was constructed based on a standard test case, and the experimental results were obtained on a computer.

[0013] First, if activity i∈N is the immediate predecessor of activity j, it means that activity j can only start after activity i is completed; the duration distribution of activity j∈N is known, with a mean of d j , the demand for the k∈R type of resources r jk , the total amount of resources of the k∈R class is R k , R is the total number of available resources.

[0014] First, let E represent the set of activity choices, and each activity choice e∈E corresponds to a triggering activity a e ∈N, corresponding to an optional activity set When the project is executed to trigger activity a e When the decision maker needs to choose from the optional activity set C e Select an activity to execute; if the trigger activity a e If it is determined not to be executed, then the optional activity set Ce All activities in will not be executed;

[0015] Let α j represents the set of dependent activities of optional activity j. If activity j is executed, then α j All activities in will be executed; otherwise α j All activities in will not be executed; let j∈C e Contains activity a e Preceded by activity j, i.e. a e <j; let j'∈α j It implies that activity j precedes activity j', that is, j < j'.

[0016] Prioritize activity d j and activities j is the construction period and quality inspection time of the original activity j, and represents the duration and quality inspection time of the kth rework, p k is the probability of rework at the kth time, and the maximum number of reworks for activity j is L j ; The mean remaining duration of activity j can be obtained The formula is expressed as:

[0017]

[0018] Prioritized,activity selection priority rules include two rules: total weighted completion time with slack time constraint and total weighted completion time with additional slack time constraint.

[0019] The activity sequencing priority rules consist of eighteen rules in total, including: fourteen rules with better performance selected from thirty-seven classic priority rules, the latest completion time rule with better performance under random RCPSP, the worst-case relaxation rule and the latest start time rule with better performance under deterministic RCPSP, and random rules.

[0020] The technical solution provided by the present invention can have the following beneficial effects:

[0021] This paper takes the flexible project scheduling problem with random duration and random rework as the research object, summarizes the classic priority rules applicable to the activity selection and activity sorting sub-problems, constructs test cases covering various scheduling environments based on standard examples, and compares and analyzes the performance of single rules and pairing rules through a large number of simulation experiments. The results show that pairing rules based on different rule combinations are significantly better than single rules. The main reason is that the classic sorting rules cannot solve the activity selection sub-problem well. In terms of the performance of pairing rules, TTSL is the selection rule with the best overall performance. This paper finds that the best performing sorting rules are the classic LFT and LST, while the optimal sorting rule RPW performs relatively average. This shows that sorting rules are more susceptible to the problem environment than selection rules. This paper finds that resource intensity and project flexibility are the two main factors affecting rule performance, while network complexity, duration variance, rework intensity and resource factors have little impact on the rules.

[0022] In summary, the present invention effectively improves the conclusion on the performance of priority rules in solving flexible project scheduling problems, and provides a more comprehensive reference for selecting appropriate priority rules for actual project scheduling decisions. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 is a schematic diagram of an example of a flexible project network according to an embodiment of the present invention;

[0024] Figure 2 This is a schematic diagram of an activity execution process with a maximum of three reworks according to an embodiment of the present invention;

[0025] Figure 3 1 is a schematic diagram comparing RDI 95% confidence intervals for ranking rules for pairing different project flexibility with TTSL according to an embodiment of the present invention;

[0026] Figure 4 1 is a schematic diagram comparing RDI 95% confidence intervals for sorting rules for pairing different construction period variances with TTSL according to an embodiment of the present invention;

[0027] Figure 5 1 is a schematic diagram comparing RDI 95% confidence intervals for sorting rules for pairing different rework intensities with TTSL according to an embodiment of the present invention;

[0028] Figure 6 1 is a schematic diagram comparing RDI 95% confidence intervals of sorting rules for pairing different network complexities with TTSL according to an embodiment of the present invention;

[0029] Figure 7 1 is a schematic diagram comparing RDI 95% confidence intervals for sorting rules for pairing different resource intensities with TTSLs according to an embodiment of the present invention;

[0030] Figure 8 1 is a schematic diagram comparing RDI 95% confidence intervals for sorting rules for pairing different resource coefficients with TTSLs according to an embodiment of the present invention; DETAILED DESCRIPTION

[0031] The embodiments of the present invention are described in detail below. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be understood as limiting the present invention.

[0032] The following combination Figures 1 to 8 , describes a priority rule method for stochastic flexible project scheduling problem according to an embodiment of the present invention.

[0033] A priority rule method for stochastic flexible project scheduling problem includes the following steps:

[0034] For resource-constrained project scheduling problems, projects are usually represented by a single-code network graph G(N,A), where N = {1, 2, ..., n} is the set of activities in the project and A represents the precedence constraint relationship between activities.

[0035] When using the priority rule to select and sort the remaining activities, the activity duration information is used in two ways: for activities without rework, the average duration is directly used for priority calculation; for activities with rework, the remaining duration of the activities with rework is estimated by aggregating the remaining duration and then the priority is calculated;

[0036] A method based on priority rules is used to solve the stochastic flexible project scheduling problem in resource-constrained project scheduling. The priority rules include activity selection priority rules and activity sorting priority rules.

[0037] In order to compare and analyze the performance of the aforementioned priority rules in solving the stochastic flexible resource-constrained project scheduling problem, a case covering different scheduling environments was constructed based on a standard test case, and the experimental results were obtained on a computer.

[0038] If activity i∈N is the immediate predecessor of activity j, it means that activity j can only start after activity i is completed; the duration distribution of activity j∈N is known, with a mean of d j , the demand for the k∈R type of resources r jk , the total amount of resources of the k∈R class is R k , R is the total number of available resources.

[0039] In addition to the activities that must be performed in the project, there is another type of activity that requires managers to make decisions and then determine whether to perform it based on the selection and dependency relationships. Let E represent the set of activity choices, and each activity choice e∈E corresponds to a triggering activity a e ∈N, corresponding to an optional activity set When the project is executed to trigger activity a e When the decision maker needs to choose from the optional activity set C e Select an activity to execute; if the trigger activity a e If it is determined not to be executed, then the optional activity set C e All activities in will not be executed;

[0040] In addition to the above opposition relationships, there may also be dependencies between optional activities. Let α j represents the set of dependent activities of optional activity j. If activity j is executed, then α j All activities in will be executed; otherwise α j All activities in will not be executed; in engineering practice, the triggering activity is usually the direct or indirect predecessor activity of the optional activity, and an optional activity is usually the direct or indirect successor activity of its dependent activity. Therefore, in order to ensure the topological order between activities, let j∈C e Contains activity a e Preceded by activity j, i.e. a e <j; let j'∈α j It implies that activity j precedes activity j', that is, j < j'.

[0041] Figure 1 This example shows a simple flexible project network. In this project, activities 1, 2, 3, 6, and 10 must be executed. The two trigger activities a1 = 1 and a2 = 5 correspond to the optional activity sets C1 = {4, 5} and C2 = {8, 9}, respectively. Furthermore, activity 4 corresponds to an associated activity set α4 = {7}. It can be seen that nested relationships between activity choices exist, meaning that a trigger activity exists within another optional activity set, such as trigger activity 5, which exists within optional activity set C1. If, when executing trigger activity 1, activity 4 is selected from C1 for execution, activities 5, 8, and 9 will not be executed, but activity 7 must be executed. Conversely, if activity 5 is selected for execution, activities 4 and 7 will not be executed, and when executing trigger activity 5, it is necessary to select between activities 8 and 9 for execution.

[0042] In addition to the aforementioned random duration and flexible network characteristics, we also consider the random rework associated with some activities. Specifically, some activities have high quality requirements and require quality inspection after completion. The inspection results must either pass or be reworked. An activity may require multiple reworks to pass, but the number of reworks is typically limited due to factors such as project cost. Activity quality inspection is typically handled by the client or other non-critical resources, so while this process takes time, it does not consume project resources. Figure 2This example illustrates the rework process for an activity, with a maximum of three reworks and probabilities of 0.3, 0.2, and 0.1, respectively. After the third rework, the activity is considered to have met quality standards, eliminating the need for further quality inspection and rework. Similarly to the original activity duration, the probability distributions of the quality inspection and rework durations are also assumed to be known.

[0043] When using priority rules to select and sort the remaining activities, activity duration information may be needed. For activities without rework, the average duration can be used directly for priority calculation. However, for activities with rework, since the number of reworks is uncertain, it is necessary to estimate the remaining duration first. When studying the parallel machine scheduling problem with random rework, an aggregated remaining duration estimation method is proposed, and it is verified that the method based on aggregated remaining duration performs better than the method based on dispersed duration estimation. Let activity d j and activities j is the construction period and quality inspection time of the original activity j, and represents the duration and quality inspection time of the kth rework, p k is the probability of rework at the kth time, and the maximum number of reworks for activity j is L j ; The mean remaining duration of activity j can be obtained The formula is expressed as:

[0044]

[0045] At the initial moment, based on the mean (remaining) duration of all activities, the forward and backward calculations of the critical path method are used to obtain the earliest (latest) start (finish) time of each activity. During the project execution, the remaining duration of the rework activity is updated using formula (1) and its priority is calculated accordingly. Figure 2 For example, the activity shown in the figure has an average remaining duration at the initial moment of When it is the first rework, the mean remaining duration is updated to

[0046] When a project reaches a triggering activity, it must select one from a set of available activities. Theoretically, the optimal activity selection should minimize the duration of the remaining project network. However, the existence of random durations, random rework, and resource constraints makes it difficult to assess which decision will achieve the optimal goal. Therefore, while existing classical priority rules can be used for activity selection, they often underperform specific priority rules for activity selection because they fail to account for the characteristics of flexible project networks.

[0047] For RCPSP-AS, two rules—the sum of durations (SOD) and the minimum total work content (TWC)—are proposed to generate the initial solution for its tabu search algorithm. The SOD rule selects the alternative activity with the smallest sum of durations among all alternative activities in the current subgraph and its branch. If nested choices exist in the remaining project network, the average duration of these alternative subgraphs is used. The TWC rule shares the same logic as the SOD rule, replacing durations with work content (i.e., the activity's resource requirements multiplied by its duration). Furthermore, seven new activity selection rules are proposed, including the Total Time to Link Subgraphs (TTSL) and Total Time to Extend Subgraphs (TTXSL) rules, which are similar to the SOD rule. The main difference between the three rules is that, for triggered but uncertainly executed optional activities and their dependent activities, the SOD rule recursively takes the mean duration of these activities, while the TTXSL rule takes the sum of their durations, while the TTSL rule does not consider these activities. Drawing on the ideas of TTSL and TTXSL rules, two new rules, TWCSL and TWCXSL, are proposed, which replaces the construction period with work content.

[0048] Activity selection priority rules include two types of rules: total weighted completion time with slack time limit and total weighted completion time with additional slack time limit.

[0049] by Figure 1 The project shown in the figure is used as an example to illustrate the calculation process of the activity selection rule. Assume that the duration settings of the optional activities are as shown in Table 2, where only activity 4 has rework. The rework process is as follows: Figure 2 As shown. Consider the first activity selection decision, the priorities of optional activities 4 and 5 need to be calculated separately. Since activity 4 has no nested choices, TTSL = TTSL = SOD = 12.18 + 4 = 16.18, TWC = 12.18 * 1 + 4 * 2 = 20.18. Activity 5 is nested, so SOD = 6 + (5 + 3) / 2 = 10, TTSL = 6, TTXSL = 6 + 5 + 3 = 14, TWC = 6 * 3 + (5 * 2 + 3 * 3) / 2 = 27.5. If SOD is used as the activity selection rule, then activity 5 has a higher priority than activity 4 and will be selected for execution. Similarly, assuming the TWC rule is used for activity selection, then activity 4 will be selected for execution.

[0050] Table 2 Figure 1 Flexible project parameter settings

[0051]

[0052]

[0053] There are eighteen activity sequencing priority rules in total, including: fourteen rules with better performance selected from thirty-seven classic priority rules, the latest completion time rule with better performance under random RCPSP, the worst-case relaxation rule and the latest start time rule with better performance under deterministic RCPSP, and random rules.

[0054] Although the priority rules for activity selection can be used for activity sorting at the same time, the combination of different priority rules performs significantly better than a single rule. Therefore, the present invention will summarize the classic priority rules that can be used for activity sorting separately.

[0055] The performance of 37 classic priority rules in solving RCPSP-AS was compared, and it was found that the least rank positional weight (LRPW) and the smallest cumulative resource requirement (SCRR) performed best under a single rule. The present invention selected 14 rules with better performance from these 37 priority rules. Considering that the random characteristics of the problem may affect the performance of the rules, the latest finish time (LFT) rule that performs better under random RCPSP was also selected. In addition, the worst case slack (WCS) and latest start time (LST) rules that perform better under deterministic RCPSP were also selected. Finally, the random rule Rand was also included in the comparison. The above 18 classic priority rules suitable for activity sorting are summarized in Table 3.

[0056] Table 3 18 classic priority rules for activity sorting

[0057]

[0058]

[0059] in:

[0060] EF i and LF i are the earliest and latest completion times of activity i.

[0061] ES i and LS i The earliest and latest start times for activity i.

[0062] AP t is the set of all waiting activities at time t, that is, the set of activities that meet the immediate predecessor constraint but have not yet started;

[0063] E(j,i) is the earliest possible start time of activity j when activity i starts at time t;

[0064] IS i and TS i are the number of immediate successor activities and the number of all subsequent activities of i respectively;

[0065] IP i is the number of immediate predecessor activities of i;

[0066] N * is the number of unscheduled activities.

[0067] level i The number of activities in the longest path from activity i to the end of the project.

[0068] w is a weight coefficient, 0≤w≤1, and the present invention takes 0.5 according to existing literature.

[0069] Experimental data

[0070] This paper reconstructs a randomized flexible RCPSP example based on the J30 example in PSPLIB. The J30 example in PSPLIB considers three problem characteristic indicators: network complexity (NC = {1.5, 1.8, 2.1}), resource coefficient (RF = {0.25, 0.5, 0.75, 1}), and resource intensity (RS = {0.2, 0.5, 0.7, 1}). Ten examples are generated for each parameter combination, for a total of 480 examples.

[0071] For each original example, the present invention also sets multiple project flexibilities, duration variances, and rework intensity. The two project flexibilities are set according to Table 4, and the activity durations are uniformly expressed using Beta distribution (referring to existing literature on random RCPSP, the duration value range is [d / 2, 2d], and the two variances are d / 3 and d 2 / 3). The rework intensity is represented by the parameter β∈(0,1). A larger β value indicates a larger rework task list, number of reworks, and rework probability. This paper considers four rework environments (β={0.2, 0.4, 0.6, 0.8}) and sets them according to Table 5.

[0072] Table 4 Definition of project flexibility

[0073]

[0074] Table 5 Rework environment settings

[0075]

[0076] To compare the relative performance of different rules, this paper uses the relative deviation index (RDI) shown in formula (2) for result analysis. The advantage of this index is that it ensures that all results fall within the interval [0, 1] and can amplify the gaps between methods, making it easier to observe subtle differences between methods.

[0077]

[0078] Among them, Cur mh Indicates the target mean obtained by solving example h using method m, Best mh and Worst mh They represent the best and worst objective values ​​obtained by all methods for solving example h.

[0079] Priority rule-based methods usually need to be combined with a specific scheduling generation mechanism to obtain a specific scheduling solution. The existing scheduling generation mechanisms mainly include serial scheduling and parallel scheduling. Among them, the parallel scheduling method can be directly applied to random RCPSP, while random serial scheduling is logically different from deterministic serial scheduling and requires that the activity sequence and start time are strictly consistent. In view of the ease of use of parallel scheduling and its good performance under complex random RCPSP, existing literature on random RCPSP usually adopts parallel scheduling methods. For this reason, the present invention also uniformly combines all priority rules with parallel scheduling methods to solve the constructed test case.

[0080] Specifically, each priority rule was simulated and solved for each case 1000 times, and the average of the results was taken for analysis. First, the performance of a single rule was tested, and the results are shown in Table 6. It can be seen that the LRPW rule has the best overall performance. However, rules such as LFT and RSM, which have similar performance to the LRPW rule, did not perform well in the experiment. Research shows that rules such as LFT actually perform well in the random RCPSP environment. Therefore, based on the above results, it can be concluded that although the performance of rules such as LFT is easily affected by the scheduling environment, the performance of the LRPW rule remains robust after adding random duration and random rework factors, which also shows that it is suitable for flexible project scheduling problems. In addition, by analyzing the changes in RDI of each rule under different feature values, it can be seen that resource intensity has the most significant impact on the rules, followed by the impact of project flexibility, and the impact of other factors is relatively small.

[0081] Table 6 Comparison of average RDI of single rules

[0082]

[0083]

[0084] Table 6 (Continued) Comparison of average RDI for a single rule

[0085]

[0086] In summary, combinations of different priority rules significantly outperform a single rule. To verify this conclusion, we combined the six activity selection rules from the activity selection priority rules with the 18 activity ordering rules shown in Table 3, resulting in 108 pairing rules. We solved all examples using each pairing rule and compared them with the results of a single rule, yielding the results shown in Table 7.

[0087] Table 7 Comparison of average RDI of pairing rules

[0088]

[0089] Table 7 shows that pairing rules significantly outperform single rules. Taking the best-performing single and pairing rules as an example, the average RDI corresponding to the best-performing pairing rule, TTSL-LFT, is 30% lower than the average RDI of the best-performing single rules, LRPW and LFT. The reason for this result is simple: appropriate rules must be used for both activity selection and activity sorting; no single rule can handle both subproblems simultaneously. As mentioned earlier, the requirements for activity selection rules are more stringent because the impact of activity selection decisions on the remaining network must be considered. The results in Table 7 show that TTSL is the best overall performing activity selection rule. However, the optimal pairing rule is TTSL-RPW, but the results in Table 7 indicate that this pairing rule performs worse than the combination of TTSL with the LFT and LST rules. This also indirectly confirms the conclusion in existing literature that LFT and LST are among the best-performing rules for solving traditional RCPSP.

[0090] To analyze the statistical significance of the performance differences between different rules, we first subjected the six activity selection rules combined with the LFT rule to a two-tailed Wilcoxon signed-rank test. The resulting P values ​​are shown in Table 8. If the P value between two rules was less than 0.05, it indicated a significant difference between the two rules; otherwise, there was no significant difference. Note that analysis of variance was not used here because the data did not meet the normality requirement for analysis of variance, which is not required by the Wilcoxon signed-rank test. The results in Table 8 show that the best-performing selection rule, TTSL, showed no significant difference only with SOD. Furthermore, there were no significant differences between SOD and TTXSL, TWC and TWCSL, or TWCSL and TWCXSL. Next, the nine sorting rules combined with TTSL were subjected to a two-tailed Wilcoxon signed-rank test. The results are shown in Table 9. The results show that the best-performing sorting rule, LFT, showed no significant difference only with LST, and there were no significant differences between the other rules.

[0091] Table 8 P values ​​of the six selection rules paired with LFT (two-tailed Wilcoxon signed-rank test)

[0092]

[0093] Table 9 P values ​​of the Wilcoxon signed rank test for the nine sorting rules paired with the TTSL rule

[0094]

[0095] Based on the test results in Table 9, we selected five sorting rules with significant differences that were paired with the TTSL selection rule to further analyze the impact of different environmental factors on their performance. The results are shown in Tables 10 and Figure 3 As shown. Similar to the results of a single rule, resource intensity still has the most significant impact on the rule, followed by the impact of project flexibility, and the impact of other factors is relatively small. Note that as mentioned earlier, RDI reflects the relative performance between different rules, rather than the absolute performance compared to the optimal result. Therefore, when the RDI of a certain method decreases, it only means that the performance of the rule is better than other rules, but in fact the absolute performance of the rule may be worse. For example, in theory, under higher variance of the construction period or higher rework intensity, the estimation error of the construction period should be larger, and the corresponding rule performance will also deteriorate. However, when the absolute performance of all rules deteriorates overall, the conclusion reflected by RDI is that the performance of each rule is relatively stable. Despite this, the relative performance of the rules reflected by RDI is still valuable for selecting reasonable decision rules. Table 10 and Figure 3It shows that when combined with the best-performing TTSL selection rule, LFT (and LST) are the relatively best-performing sorting rules in most cases.

[0096] Table 10 Comparison of average RDI of sorting rules paired with TTSL

[0097]

[0098] Table 10 (Continued) Comparison of average RDI of sorting rules paired with TTSL

[0099]

[0100] Throughout this specification, reference to terms such as "embodiment" or "example" indicates that the specific features, structures, materials, or characteristics described in conjunction with that embodiment or example are included in at least one embodiment or example of the present invention. In this specification, schematic representations of these terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

[0101] While embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions, and variations may be made to the embodiments without departing from the principles and spirit of the invention, and that the scope of the invention is defined by the claims and their equivalents.

Claims

1. A priority rule method for stochastic flexible project scheduling problem, characterized by: The steps include: For resource-constrained project scheduling problems, projects are usually represented by a single-code network graph G(N,A), where N = {1, 2, ..., n} is the set of project activities and A represents the precedence constraint relationship between activities. When using the priority rule to select and sort the remaining activities, the activity duration information is used in two ways: for activities without rework, the average duration is directly used for priority calculation; for activities with rework, the remaining duration of the activities with rework is estimated by aggregating the remaining duration and then the priority is calculated; A method based on priority rules is used to solve the stochastic flexible project scheduling problem in resource-constrained project scheduling. The priority rules include activity selection priority rules and activity sorting priority rules. In order to compare and analyze the performance of the aforementioned priority rules in solving the stochastic flexible resource-constrained project scheduling problem, a case covering different scheduling environments was constructed based on a standard test case, and the experimental results were obtained on a computer.

2. A priority rule method for stochastic flexible project scheduling problem according to claim 1, characterized in that: If activity i∈N is the immediate predecessor of activity j, it means that activity j can only be started after activity i is completed; The duration distribution of activity j∈N is known, and its mean is d j , the demand for the k∈R type of resources r jk , the total amount of resources of the k∈R class is R k , R is the total number of available resources.

3. The priority rule method for stochastic flexible project scheduling problem according to claim 1 is characterized in that: Let E represent the set of activity choices, each activity choice e∈E corresponds to a triggering activity a e ∈N, corresponding to an optional activity set When the project is executed to trigger activity a e When the decision maker needs to choose from the optional activity set C e Select an activity to execute; if the trigger activity a e If it is determined not to be executed, then the optional activity set C e All activities in will not be executed; Let α j represents the set of dependent activities of optional activity j. If activity j is executed, then α j All activities in will be executed; otherwise α j All activities in will not be executed; let j∈C e Contains activity a e Preceded by activity j, i.e. a e <j; let j'∈α j It implies that activity j precedes activity j', that is, j < j'.

4. The priority rule method for stochastic flexible project scheduling problem according to claim 1 is characterized in that: Order activity d j and activities j is the construction period and quality inspection time of the original activity j, and represents the duration and quality inspection time of the kth rework, p k is the probability of rework at the kth time, and the maximum number of reworks for activity j is L j ; The mean remaining duration of activity j can be obtained The formula is expressed as:

5. The priority rule method for stochastic flexible project scheduling problem according to claim 1 is characterized in that: Activity selection priority rules include two types of rules: total weighted completion time with slack time limit and total weighted completion time with additional slack time limit.

6. The priority rule method for stochastic flexible project scheduling problem according to claim 1 is characterized in that: There are eighteen activity sequencing priority rules in total, including: fourteen rules with better performance selected from thirty-seven classic priority rules, the latest completion time rule with better performance under random RCPSP, the worst-case relaxation rule and the latest start time rule with better performance under deterministic RCPSP, and random rules.

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