This paper focuses on a multiproject resource allocation problem in a bilevel organization. To solve this problem, a bilevel multiproject resource allocation model under a fuzzy random environment is proposed. Two levels of decision makers are considered in the model. On the upper level, the company manager aims to allocate the company's resources to multiple projects to achieve the lowest cost, which include resource costs and a tardiness penalty. On the lower level, each project manager attempts to schedule their resource-constrained project, with minimization of project duration as the main objective. In contrast to prior studies, uncertainty in resource allocation has been explicitly considered. Specifically, our research uses fuzzy random variables to model uncertain activity durations and resource costs. To search for the optimal solution of the bilevel model, a hybrid algorithm made up of an adaptive particle swarm optimization, an adaptive hybrid genetic algorithm, and a fuzzy random simulation algorithm is also proposed. Finally, the efficiency of the proposed model and algorithm is evaluated through a practical case from an industrial equipment installation company. The results show that the proposed model is efficient in dealing with practical resource allocation problems in a bilevel organization.
Because more and more construction companies must deal with multiple projects at the same time, both the theory and practice of multiproject resource allocation problems (MPRAP) are being paid increasing attention in the construction industry. In existing researches, resource allocation has often been considered only a constraint in multiple project scheduling problems and thus MPRAP has often been called a resource-constrained multiple project scheduling problem [
All this research has assisted in the improvement of multiple project resource allocation. However, it is still commonly assumed that a single manager oversees all projects. In today’s industrial climate, managers face an increasingly complicated decision environment. As a result, a single manager has difficulties in dealing with across project resource allocation in addition to resource management within projects. In this case, a bilevel organizational structure is frequently used for project management [
In addition to the complexities of the bilevel structure, uncertainty is also frequently considered in resource allocation problems. Methods for dealing with uncertainty in decision-making mainly include random, fuzzy, and interval mathematical programming [
Hence, this paper focuses on this type of bilevel multiproject resource allocation problem (BLMPRAP) under a fuzzy random environment, in which we attempt to find an optimal allocation scheme using a bilevel programming model. In this model, the decision maker on the upper level is the company manager whose aim is to determine an optimal scheme for the allocation of company resource among multiple projects. The objective of the company manager is to minimize total cost which consists of resource cost and a tardiness penalty, while at the same time considering the lower-level decision-making. On the lower level, each project manager attempts to schedule their project in the most efficient way using the assigned resources. Contrary to the company manager, the project manager’s objectives are focused on the project duration and finishing time of the project, rather than the cost. Therefore, the minimization of project duration is considered the objective on the lower level. In addition, the uncertainty associated with activity duration and resource costs is also explicitly considered in the model. Specifically, our research uses fuzzy random variables to model the activity duration and resource costs. Moreover, we also focus on a solution method for the proposed bilevel resource allocation model and two main heuristic methods are discussed in the algorithm section, and a solution method which integrates these two algorithms is proposed. Finally, a representative case is used to test the model and algorithm.
The remainder of this paper is organized as follows. In Section
The problem considered in this paper is a bilevel resource allocation problem with multiple projects under a fuzzy random environment. In this section we explain why this problem should be solved using a bilevel programming model and outline the procedure for modeling uncertain activity duration and resource cost using fuzzy random variables.
In practice, more and more companies are concurrently managing multiple projects with limited resources. In order to service each project better, a hierarchical (bilevel) organizational structure which consists of a company level and a project level is being used by many companies such as construction companies, software companies, and some production companies. In these cases, the company managers need to deal with hierarchical decision-making.
To handle these decentralized optimal planning problems in a hierarchical (or multiple-level) organization which has more than one decision maker, multilevel mathematical programming has been proposed [
In this paper, resource allocation is considered across multiple projects in a bilevel organization which includes two levels of managers (i.e., company managers and project managers). On the upper level, the company managers are generally responsible for corporate planning and coordination between the multiple project groups with the aim of maximizing the company’s income. In the construction industry, they generally control and manage key company resources, such as large-scale equipment and senior engineering staff. However, resources are generally limited and some are also very expensive. To save costs, company managers have to make detailed resource assignment plans over multiple projects. Cost is dependent on the practical project schedule as a tardiness penalty occurs if the project duration exceeds its contracted finishing time, so company managers must also consider the project managers’ decisions when planning their resource allocation over multiple projects. On the lower level, project managers also maintain a reasonable level of resources called the “project resource.” The manager of each project is responsible for resource allocation (including project resources and assigned company resources) over multiple project activities to ensure that the project is completed on time, so they also have to develop a resource-constrained project schedule after the company resources assignment plan has been completed.
Usually, there are different objectives between the company’s projects and the project managers. The company managers desire a resources assignment plan that achieves a lower cost and a shorter duration. However, at the same time, the achievement of these objectives is dependent on not only the upper-level decision-making, but also the actions of the project managers. The project managers pay more attention to the finishing time or the cost of a single project, which may be in contradiction to the company’s benefit. Company managers know that the project managers make decisions based on the assigned company resources, so to some degree they have some influence on the project managers’ decisions through the different resource allocation schemes. Therefore, the considered multiproject resource allocation is a decision-making problem in a bilevel organization with a degree of conflict in terms of benefits. It is appropriate to solve this problem using bilevel programming. In bilevel programming, the decision maker on the upper level is the company manager who seeks to allocate company resources to multiple projects at the lowest cost. On the lower level, each project manager attempts to schedule their project with the objective of project duration minimization under resource constraints. The bilevel resource assignment problem is illustrated in Figure
Flow chart for the considered bilevel resource assignment problem.
The fuzzy random environment has been studied and applied to many areas such as inventory problem [
In real conditions, uncertainty analysis is always an important consideration for managers in many areas of operations, such as the uncertainty that exists in activity durations, resource requirements, and operating costs. In this paper, our main consideration is project activity duration and unit resource cost uncertainty.
Activity durations are always uncertain because of a lack of knowledge and in previous studies they have often modeled these uncertainties as random or fuzzy variables. However, there are often circumstances where both fuzzy and random factors exist in a complex uncertain environment. For example, a company plans to install a boiler in a power plant construction project in October, but they do not have enough experience or historical data on this type of project. In this case, fuzzy variables are used to model the activity durations. At the same time, some known information associated with the activity duration, such as the effects of the weather, can be modeled as a random variable. For example, a shower may slow down the transportation speed of some necessary equipment or extreme temperatures may lead to lower work efficiency. From the local statistical information, in October, it is predicted with a probability of 0.6 to rain, with a probability of 0.3 to be fine, and a probability of 0.1 to be cloudy. Therefore, the weather can be modelled as a discrete random variable. In this situation, activity durations considering both fuzzy factors and random factors can be modelled as fuzzy random variables as shown in Figure
Employing fuzzy random variables to model activity durations.
The situation is similar for resource costs. For example, as the gasoline price and the crane operators’ wages are expected to rise, the cost of crane operations will also go up. However, it is very difficult to obtain a precise value because of the many uncertainties. In this case, an interval
Employing a fuzzy random variable to express the unit cost for company resources.
Considering the bilevel structure and uncertain environment simultaneously, the BLMPRAP under a fuzzy random environment can be stated as follows. A company has contracted n projects at the same time, though the company managers are unable to fully manage these projects, so for effective management they take charge of only some key resources and establish project groups to manage the projects. The problem the company managers face is how to assign the company resources to each project group, while the project manager has to schedule their project with some resource constraints. To deal with this uncertainty, the activity durations and resource costs are modeled as fuzzy random variables. The decision-making framework for the proposed bilevel multiproject resource assignment problem is illustrated in Figure
Decision-making framework for the considered BLMPRAP.
To solve the multiproject resource allocation problem in a bilevel organization, a bilevel programming model under a fuzzy random environment is constructed. The mathematical description for this problem is given as follows.
To model the problem more efficiently, the following assumptions are adopted. The bilevel resource assignment problem consists of multiple resources and multiple projects. There are no new projects during the scheduled resource allocation periods. The problem has two levels of decision makers, that is, company managers on the upper level and project managers on the lower level. The managerial objective on the upper level is to minimize the total cost for all projects, and the objective on the lower level is to minimize the project duration. A single project consists of a number of activities each with several optional execution modes. Each mode is a combination of duration and resource requirements [ Each activity needs multiple types of resources. The unit cost for each company resource and the duration for each activity are modelled as fuzzy random variables. The company manager is responsible for resource allocation during multiple projects. Resources assigned to all projects do not exceed the limited quantities in any time period.
The problem the company manager on the upper level faces is how to allocate the limited company resources over several projects in each period (generally the period is one week); in other words, they need to decide the quantity to be allocated to each project in each period for each type of resource. With this in mind, the decision variables for the upper level are
For resource allocation problems, minimization of the total cost or maximization of the total profit is often considered as the decision objective [
Therefore, the total tardiness penalty can be stated as
In this equation, the unit resource cost
In order to introduce the chance-constrained programming, the concept of a chance measure for the fuzzy random variables is first explained. Let
From the definition of the chance measure, we can derive the following equation:
Since it is not possible to derive a precise minimum objective, the decision makers descend to seek a minimum objective value
Resource constraints must be met for all types of resource allocation problems. That is, for each type of resource, the total quantity allocated to every project cannot exceed the ownership quantity of the company in each period. This constraint is described as
Equations ((
When the resources are allocated to each project, the project manager has to consider how to make use of these resources to finish the project more quickly. Therefore, each project manager is faced with a resource-constrained project scheduling problem. Usually, the resources consist of company resources and project resources. In this paper, we only consider the company resources when scheduling the project.
For project scheduling, the minimization of project duration is often considered as the decision objective [
In addition, some constraints must be met. First, each activity must be scheduled and its finish time must be in the range of its earliest finishing time and its latest possible finishing while ensuring that all activities are adequately arranged and there is only one execution mode for each activity. So we can get the following constraint:
In the scheduling problem, precedence is the basic term which ensures the rationality of the arrangement.
The fuzzy expected value reflects the center value that the fuzzy random variable tends towards and describes the fuzzy random variable statistical properties. After going through the fuzzy expected operation above, all fuzzy random durations are transformed into fuzzy durations. Then the expected value operator of the fuzzy variables based on a fuzzy measure [
From the fuzzy random expected value operator and the fuzzy expected value operator, the expected precedence constraints can be obtained as
In addition to the precedence constraints, resource constraints must be considered as well in this problem. In each period, the available resource quantity is
The objective function and the constraints form the resource-constrained project scheduling model as in
There are two levels of decision makers in the considered BLMPRAP. The decision maker on the upper level, the company manager, hopes to allocate the company resources to multiple projects at the lowest cost. The cost consists of resource costs and the tardiness penalty, so the upper-level decision maker is able to control the resource cost through appropriate allocation. The tardiness penalty is dependent on the finishing time of all projects, which in turn is determined by the specific project managers through their project schedule. In this situation, the company manager must consider the decision of the project managers. The company manager does know that the project managers must schedule their projects under the resource constraints. Therefore, the company manager can influence the decision-making of project managers on the lower-level model using different resource allocation schemes.
On the lower level, each project manager attempts to make a more efficient schedule under the resource constraints. The objective is often to minimize the finishing time of the project, although this may conflict with the company’s objective. This is another reason why such a problem needs to be modeled as a bilevel programming model. In addition, uncertainty also impacts the decision. In this paper, the uncertain resource cost and activity duration are described using fuzzy random variables. On the upper level, possibility theory is used to deal with the uncertain resource cost. On the lower level, an expected value operation is used to cope with the uncertain activity duration. In sum, the complete bilevel programming model can be established based on the above discussion as in (
The proposed model is a bilevel programming model, which is considered as a strong NP-hard problem [
To solve the bilevel model, a particle swarm optimization is proposed to search for the solution to the upper level. At the beginning of the algorithm, some feasible solutions (particles:
The overall procedure of the proposed solution algorithm.
To solve the bilevel model, an improved adaptive PSO is introduced to cope with the upper-level programming. In contrast to classical PSO, to improve the convergence speed, a float coding method, which is capable of incorporating various constraints in its implementation [
Set the parameters for the adaptive PSO: swarm_size, iteration_max,
Initialize the velocity and the position of the upper-level model. Each particle is represented as
Solve the lower-level programming with the initialization result of the upper-level variables using the proposed adaptive hybrid genetic algorithm for the multimode resource-constrained project scheduling problem.
Calculate each fitness value using a fuzzy random simulation procedure using the calculated results of the lower level:
Step 1: Let Step 2: Generate Step 3: Generate a determined vector Step 4: If Step 5: If
Update the
Update the inertia weight for iteration
Update the velocity and the position of each particle using the following equations:
If the stopping criterion is met, that is,
In the considered problem, the multimode resource-constrained project scheduling problem (MRCPSP) is discussed on the lower level. For the MRCPSP, many types of heuristic algorithms such as simulated annealing [
Set the initial value and parameters for the genetic algorithm: population size, crossover rate
Generate the initial population
An individual solution composed of priority-based and multistage-based chromosomes.
Evaluate
Create
Order-based crossover for activity priority.
Create
Climb
Apply the heuristic for adaptively regulating GA parameters. Select
Here, the
Repeat the above stages 3 to 7 after
In this section, computational experiments that were carried out on a real application are presented. Through an illustrative example on the data set adopted from a case study, the proposed method is validated and the efficiency of the algorithm is tested. The data for resource allocation, project scheduling, and others involved in the case are from an industrial equipment installation company (company X) and an electric power design institute in Sichuan province, China. The case is introduced to demonstrate the potential real world applications of the proposed methods.
Company X is a state-owned large-scale comprehensive installation and construction company with total assets of 460 million RMB and more than 3000 workers, which always contracts for multiple projects at the same time. To manage these projects, many project groups are found. These project groups can purchase some materials and equipment by themselves. However, some other resources must be allocated from the company such as large-scale equipment and professional staff.
The company has contracted for an installation engineering project at the HP power plant construction project in Luzhou. This is made up of two projects: the installation projects of 1 and 2 power units. At the same time, the company has also contracted for another installation project: an equipment installation project for a sewage treatment construction engineering project in Luzhou. Hence, the company is managing three projects at the same time. For management convenience, each project is managed by a project group who takes charge of the project scheduling and resource allocation within their project. However, some important resources such as large-scale installation equipment are still controlled by the company manager. The problem the company manager faces is how to allocate these resources over the three projects so as to gain maximal company income. This is a good example of the proposed bilevel resource allocation problem.
In this case, each power plant construction engineering installation project consists of 12 activities, while the sewage treatment construction engineering equipment installation project has 11 activities. The flow charts are illustrated in Figures
The activity duration and resource consumption for the installation projects.
No. | Mode | Duration | Resource requirement | |||
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CR | CM | WO | EE | |||
16 | 10 | 30 | 6 | |||
A | 1 |
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6 | 4 | 8 | 2 |
2 |
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8 | 5 | 12 | 2 | |
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B | 1 |
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3 | 2 | 4 | 2 |
2 |
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4 | 2 | 6 | 2 | |
3 |
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5 | 4 | 8 | 2 | |
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C | 1 |
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2 | 3 | 0 | 0 |
2 |
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3 | 3 | 0 | 0 | |
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D | 1 |
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2 | 0 | 4 | 0 |
2 |
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2 | 0 | 6 | 0 | |
3 |
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3 | 0 | 8 | 0 | |
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E | 1 |
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2 | 0 | 3 | 0 |
2 |
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3 | 0 | 4 | 0 | |
3 |
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4 | 0 | 5 | 0 | |
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F | 1 |
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2 | 4 | 0 | 0 |
2 |
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3 | 6 | 0 | 0 | |
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G | 1 |
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2 | 0 | 8 | 1 |
2 |
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3 | 0 | 10 | 2 | |
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H | 1 |
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0 | 0 | 0 | 4 |
2 |
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0 | 0 | 0 | 6 | |
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I | 1 |
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2 | 2 | 4 | 0 |
2 |
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4 | 3 | 6 | 0 | |
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J | 1 |
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2 | 0 | 0 | 2 |
2 |
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2 | 0 | 0 | 2 | |
3 |
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3 | 0 | 0 | 3 | |
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K | 1 |
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1 | 0 | 3 | 1 |
2 |
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2 | 0 | 5 | 2 | |
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L | 1 |
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0 | 0 | 4 | 2 |
2 |
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0 | 0 | 6 | 3 | |
3 |
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0 | 0 | 8 | 4 |
The total quantity and unit cost of the company resources.
Resource | CR | CM | WO | EE |
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Total quantity | 16 | 10 | 30 | 6 |
Cost |
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The activity precedence of installation project for thermal power plant construction engineering.
The activity precedence of installation project for sewage treatment construction engineering.
In order to run the program for the proposed PSO-GA algorithm, the parameters for the PSO algorithm were set as follows: swarm_size=40, iteration_max=200, inertia weight_max=1, weight_min=0, position acceleration constant
The computer running environment was an intercore 2 Duo 2.26 GHz clock pulse with 2048 MB memory. The program was written using MATLAB 2007. After 3.12 minutes on average, the optimal solutions for the bilevel programming were determined.
The partial assignment scheme for these resources is shown in Table
An optimal resource assignment scheme after 20 experiments.
Time | CR | CM | WO | EE | ||||||||
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1–8 | 8 | 8 | 0 | 5 | 5 | 0 | 12 | 12 | 0 | 2 | 2 | 0 |
9–12 | 4 | 4 | 8 | 2 | 2 | 4 | 6 | 6 | 10 | 2 | 2 | 2 |
13–16 | 6 | 6 | 4 | 5 | 5 | 0 | 8 | 8 | 12 | 0 | 0 | 2 |
17–20 | 7 | 4 | 4 | 4 | 4 | 2 | 19 | 11 | 0 | 2 | 2 | 0 |
21 | 5 | 4 | 4 | 0 | 0 | 4 | 12 | 8 | 8 | 0 | 2 | 2 |
22 | 4 | 4 | 4 | 0 | 0 | 6 | 9 | 13 | 6 | 2 | 2 | 0 |
23-24 | 5 | 4 | 4 | 0 | 0 | 4 | 10 | 12 | 8 | 4 | 2 | 0 |
25-26 | 5 | 4 | 6 | 4 | 4 | 2 | 4 | 8 | 8 | 4 | 0 | 2 |
The allocation plan for Resource CR during 38 weeks.
In this section, the proposed model is analyzed through a comparison with other resource allocation methods and an analysis is given for three uncertain models.
Traditionally, resource allocation planning over multiple projects is executed using a resource-constrained multiple project scheduling model (RCMPS). This model is dependent on an assumption that a single manager oversees all projects. That is, there is only one level manager who is responsible for the overall project resource allocation and for the resource allocation in each specific project. However, a bilevel organization structure is frequently used to manage projects which have two levels of managers. In this case, a bilevel optimization assignment model (BLOAM) is proposed to allocate the company resources over multiple projects. In this model, the company managers are responsible for the multiproject resources allocation in each period, while the project managers are responsible for the resource allocation for each specific project. In practice, other several assignment methods can also be used in the bilevel multiproject environment. One of these is called the Simple Weight Allocation Method (SWAM). The SWAM gives a weigh to each project and then assigns resources to the projects according to the weight at the beginning of each resource period. Another method is the First In System First Served (FISFS) method which gives priority to the project that has been waiting the longest when resource conflicts occur. In addition, the MINPDD method gives priority to the project that has the earliest project due date and the MINPSLK method gives priority to the project that has the smallest project slack. There is a common characteristic in these four methods, as all of them only set the allocation regulation before the project implementation rather than making a detailed allocation plan for each time period.
To test these methods above, four performance measures were used: total cost, project finishing time, actual usage, and total resource transfers. Actual usage refers to the proportion of the used resources compared to the total assigned resources. In practice, in order to improve resource usage, the assigned resource is transferred to each project at the beginning of each time period. At the end, idle resources should be released back into company’s resource pool if they are not required for a project in the next time period. This resource transfer is used to record the transferred resource quantity between the company resource pool and the projects.
The computation results from the five methods are shown in Table
An optimal resource assignment scheme after 20 experiments.
Assign methods |
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Finishing time | Actual usage | Total resource | ||
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Project 1 | Project 2 | Project 3 | ||||
BLOAM | 5242 | September 6 | November 16 | September 25 | 96.42% | 224 |
SWAM | 6381 | September 6 | November 16 | September 25 | 78.38% | 136 |
FISFS | 5768 | September 6 | November 24 | October 15 | 92.13% | 248 |
MINPDD | 5966 | September 6 | November 24 | October 15 | 94.35% | 236 |
MINPSLK | 5842 | September 26 | November 24 | September 28 | 89.72% | 198 |
Uncertainty is an important consideration in this study. In particular, fuzzy random variables which integrate fuzzy factor and random factor are used to model the uncertain activity durations and resource costs because of the lack of precise data. Besides the fuzzy random variables, we can also use fuzzy variable or random variables to deal with the uncertainty. If only fuzzy factors are considered, then some important random information such as the weather has to be ignored. The situation is similar when only considering the random factors. Taking the duration of activity
Compared with only fuzzy factors, the fuzzy random duration has more information which can lead to a more precise calculation. If we only consider the random factors, then the fuzzy number in the fuzzy random number has to be replaced using a crisp number. The crisp number is chosen in its interval randomly. This could lead to unstable, even false computation results.
A test was also made by solving the proposed bilevel optimization model using the three types of uncertain data. In the comparison, the fuzzy random model and fuzzy random found different solutions by adjusting the optimistic and pessimistic index
Comparisons among the three types of uncertainty.
Type of uncertain | Best result | Worst result | Average result |
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Fuzzy random | 5240 | 5386 | 5292 |
Fuzzy | 5259 | 5421 | 5306 |
Random | 5204 | 5506 | 5318 |
In this paper, a hybrid algorithm made up of an adaptive PSO, a GA, and a fuzzy random simulation was proposed to solve a bilevel resource allocation problem. In order to test the efficiency of the algorithm, a comparison with other solution methods was conducted.
The most common solution strategy for bilevel model is to transform it into a single level using the Karush-Kuhn-Tucker (KKT) conditions. However, this is difficult when variables only take integer values in the inner models. This also means that it can not be solved using common commercial solvers. Hence, it is more appropriate to solve the problem using a heuristic algorithm. In this paper, an improved adaptive PSO was proposed to deal with the upper level model. First, a comparison of the improved aPSO and original PSO was carried out. The average convergence curves are shown in Figure
The schedules of three projects on the lower level.
The convergence curves of the three PSO algorithms.
In addition, in our problem, the lower-level model is also better to be solved using a heuristic algorithm. Traditionally, researchers have tended to use the same algorithm to solve both the upper level and lower level. However, for a multimode resource-constrained project scheduling problem, the genetic algorithm shows a significantly higher percentage of success in finding the optimal solution although it may be slower. If the optimal solution to the lower-level model cannot be found, then the final solution may not be feasible. Hence, an adaptive genetic algorithm was proposed to deal with the lower-level model in this paper. In order to test the efficiency of the proposed hybrid algorithm, other bilevel algorithms such as PSO-PSO, GA-GA, and GA-PSO were also tested over 50 experiments. In the experiments, the PSO [
Performance of the proposed algorithms based on 50 experiments.
Performance | PSO-GA | PSO-PSO | GA-PSO | GA-GA |
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Best result | 5,234,486 | 5,382,321 | 5,323,732 | 5,224,494 |
Average result | 5,318,216 | 5,547,291 | 5,388,462 | 5,267,782 |
Computing time | 3.23 | 2.15 | 5.29 | 14.84 |
This paper presented a bilevel optimization model for company resource allocation among multiple projects in a hierarchical organization. There are two levels of decision makers in the model. The decision maker on the upper level is the company manager who hopes to allocate company resource to multiple projects at a lower cost. This cost consists of the resource costs and the tardiness penalty. On the lower level, each project manager attempts to schedule their project with the objective of minimization of project duration under resource constraints and multiple modes. In addition, the uncertainty associated with activity duration and resource cost has been explicitly considered in the model. Specifically, our research used fuzzy random variables to model the activity duration and resource costs. Then a hybrid algorithm made up of an adaptive PSO and a GA based on fuzzy random simulation was also applied to search for the optimal solution to the bilevel model. In the algorithm, an adaptive PSO was introduced to cope with the upper level programming, while an adaptive hybrid genetic algorithm was embedded into the PSO to solve the lower-level model. Finally, the efficiency of the proposed model and algorithm was evaluated using a practical case and various computing attributes. In contrast to prior studies, the proposed model shows that it was able to deal with a multiproject resource allocation in a bilevel optimization such as in most of construction companies, software companies, and some production companies. The limitation of the proposed model is that it does not allow for new projects to be added during the scheduled resource allocation periods. This is an interest area for our future research. In addition, in future research we also expect to investigate additional methods for dealing with the uncertainty in resource management such as using interval mathematical programming, which has been successfully applied to environmental management [
The authors declare that they have no conflict of interests.
This research was supported by the National Science Foundation for the Key Program of NSFC (Grant no. 70831005) and “985” Program of Sichuan University “Innovative Research Base for Economic Development and Management.”