Introduction

The resource-constrained multi-project scheduling problem (RCMPSP), as a generalization of the traditional resource-constrained project scheduling problem (RCPSP) [1], is more pervasive in today’s enterprise project management. According to Turner [2], at least 90% of projects are executed in a multi-project environment. Indeed, more and more companies are likely to manage more than one project simultaneously for the purpose of lower management cost and shorter completion time. As mentioned by Payne [3], managing multiple projects at the same time could bring tremendous benefits for the business firms. Moreover, the current multi-project context is becoming more and more distributed with the speedy development of Internet technology and globalization [4], which means that the multiple projects could be located at different places and each project is managed by an autonomous decision maker (i.e., local project manager). In this circumstance, local project managers are responsible for scheduling all activities within their respective projects independently. Consequently, the distributed resource constrained multi-project scheduling problem (DRCMPSP) has been put forward formally [5].

A great deal of multi-project scheduling problems in service operations and manufacturing are managed and run in a distributed way. Examples can be commonly found in maintenance services [6,7] and supply chain management [8–10]. How to determine the right location for shared storage facilities and allocate the limited global resources to each project is vital for achieving the success of multi-project scheduling with resource transfers [11]. In the past, RCMPSP is mainly solved by centralized methods, which focus on allocating resources and scheduling activities for all projects by only one decision maker in an integrated way [12,13]. Specifically, multiple projects are combined into a single mega-project with two dummy activities which represent the start and end of all the projects respectively. Compared to the integrated means, DRCMPSP dedicates to making decisions in a distributed manner. To be more specific, all projects running parallel in different sites are performed by different decision makers i.e., the local project managers. Given the distributed nature, DRCMPSP involves the coordination of shared global resources among multiple projects. Owing to the assumption of asymmetric distribution of information in DRCMPSP, multi-agent systems (MAS) are used to make distributed decisions [14]. A MAS represents a decentralized system including a set of independent, autonomous and self-interested agents [15]. It is prevalent that companies adopt the dual level management structure for multi-project scheduling: the project management office (i.e., senior manager) who controls the resource allocation and coordination among the rival projects, and the project managers who are responsible for the specific affairs of each project such as scheduling [16]. Therefore, agents playing the roles of senior manager and project manager should be considered in MAS. Furthermore, the coordination mechanism for shared resources allocation is the key to solving DRCMPSP.

In this paper, we employ a two-stage decomposition approach based on a new MAS for DRCMPSP. It corresponds to the dual level managerial mechanism in actual enterprises. In stage one, each project manager is devoted to completing all tasks as soon as possible. In stage two, the senior manager acts as a coordinator agent and is mainly responsible for eliminating the resource conflicts among competing projects by a novel sequential game-based negotiation mechanism. Taking into account the delay penalty such as customer complaints and reputation loss, a senior manager usually attaches more importance to the total tardiness cost (TTC) rather than the most regular and popular measure of time performance (e.g. average project delay (APD)) in practical multi-project management scenarios [17]. Therefore, we develop the forward-backward hybrid genetic algorithm (FBHGA) to generate an initial local schedule with the goal of minimizing project makespan for each project manager in stage one, and design a sequential game-based negotiation mechanism to coordinate the global resources allocation for the senior manager in stage two where the best subgame perfect Nash equilibrium is determined with minimized TTC after several iterations. In order to test the proposed approach, we implement computational experiments on the instances from MPSPLIB problem library for DRCMPSP [18]. The results given by FBHGA in stage one are compared to the optimal solutions given by the branch-and-bound (B&B) algorithm embedded in RESCON [19]. To illustrate the strength of the two-stage decomposition approach with sequential game-based negotiation mechanism, we also compare our approach with two state-of-the-art decentralized methods with respect to the objective of minimizing TTC.

The remainder of this paper is structured as follows: an overview of the previous work on DRCMPSP is provided in Section 2, followed by a detailed description of the problem discussed in this paper (Section 3). In Section 4, we introduce the agent-based two-stage decomposition approach. Section 5 demonstrates an illustrative example. Section 6 explores the experiments and analyzes the computational results. Concluding remarks are drawn in the last Section 7.

Literature review

Resource-constrained project scheduling problem (RCPSP), as one of the most complicated problems of operation research, has gained significant importance in the past few years. It is a classical and well-known standard problem in project management [20]. As a generalization of the job-shop scheduling problem, it is NP-hard in the strong sense [21]. Demeulemeester & Herroelen [22] proposed a depth-first branch-and-bound algorithm to solve the exact optimal solutions for smaller instances. A large number of heuristic and meta-heuristic procedures have been developed to obtain the approximate optimal solutions for large-scale problems. Kolisch [23] presented the priority rule-based approached for RCPSP and evaluated the performance of different priority rules. Lova & Tormos [12] analyzed the effect of serial and parallel schedule generation schemes with five priority rules for single project and multi-project instances, respectively. According to the computational study, the parallel scheduling generation scheme was superior for multi-project scheduling problem. Gonçalvesabc [24] and Mendes et al. [25] employed a genetic algorithm (GA) for RCPSP where the chromosome representation was based on the random keys. Moreover, the simulated annealing (SA) [26], tabu search (TS) [27] and ant colony optimization (ACO) [28] algorithms were also widely used for RCPSP. Hartman & Kolisch [29] summarized and categorized a large of heuristics in the RCPSP literature. They also evaluated the performance of several state-of-the-art algorithms and examined the impact of problem characteristics on it. The experimental investigation was updated a few years later [30].

While a large body of public literature addresses RCPSP, less importance has been attached to the multiple projects cases. It goes without saying that the great majority of papers on multi-project scheduling problem concentrate on allocating shared resources in an integrated manner where the multiple projects are controlled by only one manager [12,31,32]. In light of the distributed environment of multi-project management in the real world, studies on DRCMPSP have attracted increasing attention in the last decade. Traditional approaches to RCMPSP are no longer applicable to DRCMPSP owing to the informational decentralization. MAS is generally used to formulate DRCMPSP. Moreover, different coordination mechanisms for global resources allocation will exert a significant effect on the quality of solutions.

Some of the existing studies are committed to market-based mechanisms. Lee et al. [4] put forward a dynamic economy model, where a market-based control mechanism was developed to allocate limited shared resources for each required project to minimize APD. Confessore et al. [5] formally distinguished between local resources and global resources. An iterative combinatorial auction mechanism was proposed to address the global resource conflicts. Unfortunately, above-stated approaches were only suitable for small problems with the assumption that all global resources were single-unit. Araúzo et al. [33] presented a combinatorial auction mechanism to allocate the limited resources for project portfolio. Only global resources were considered. Adhau et al. [34] proposed a multi-unit combinatorial auction-based negotiation approach (DMAS/ASN). To minimize APD, three cost elements were considered in calculating the bidders’ utility. The authors further extended the above model with the assumption that the cost and time in transferring global resources could not be ignored [35]. Song et al. [36] resolved the DRCMPSP based on the multi-unit combinatorial auction method and the global resource conflicts were coordinated on project level instead of activity level. In addition to the above-mentioned market-based mechanisms to address DRCMPSP, negotiation based approaches have also gained tremendous popularity. Homberger [37] proposed a MAS-based restart evolution strategy to coordinate the allocation of global resources. Several scheduling agents with a mediator agent were contained in the MAS. On the basis of this work, the author presented a generic negotiation-based mechanism to solve all the problem instances in MPSPLIB problem library with the goal of minimizing APD [18]. Wauters et al. [38] introduced a learning-based game theoretic method depending on a simple sequence learning game played by multiple project agents and coordinated by a mediator agent. The minimization of APD and total makespan (TMS) were considered as the objectives. Zheng et al. [39] proposed a critical chain based elimination mechanism to resolve resource conflicts by considering both resource constraints and precedence constraints. The DMAS/EM algorithm was developed to obtain low APD of solutions even for large-size problems.

The above literature review shows that much more attention has been paid to solve the DRCMPSP with the objective of minimizing APD. Unfortunately, according to Vanhoucke et al. [17], most enterprises often faced the tardiness penalties as a result of project completion delays when they hiring subcontractors, research teams and maintenance crews. From a more realistic point of view, the senior manager would prefer to achieve a global TTC objective of multiple projects rather than a uniformity local delay goal of each project [40]. Since each project has different delay penalties, the multi-project schedule that minimizes APD does not ensure a minimized TTC. Each project manager is more concerned about completing the project as early as possible while the senior manager takes the TTC of the multi-project schedule as his primary consideration. Rather than aiming at the objective of minimizing APD as in the literature for DRCMPSP, this paper concentrates on developing a MAS-based approach to minimize TTC, which fulfills such gaps in existing works. In addition, a group of project managers plays sequential games to compete for global resources and eliminate conflicts. The sequential game theoretic approach has gained increasing popularity in optimizing resource allocation in grid environment [41,42] and integrating production and maintenance in the permutation flow shop sequencing problem [43]. We propose a two-stage decomposition approach to DRCMPSP by applying a new MAS and a sequential game-based negotiation mechanism.

Problem description

In DRCMPSP, two types of renewable resources need to be distinguished which are required by each project. The local resources are only available to the corresponding project while the global resources are always shared by all projects. There is no relationship among projects except the shared global resources. Each individual project manager makes scheduling decision independently. The objective of the DRCMPSP proposed in this paper is to determine a multi-project schedule that minimizes TTC, such that the precedence constraints and local/global resource constraints are all satisfied. The symbols involved in this paper are defined as follows:

• There are m (i = 1, 2, …, m) projects to be scheduled simultaneously. Each project i has an arrival date ad_i≥0 representing the earliest start time [44]. For each project i, the unit tardiness cost tc_i≥0 is given [45].
• Each project i consists of J_i (j = 1, 2, …, J_i) non-preemptable activities. a_ij stands for the jth activity of project i. Two dummy activities a_i0 and are added to each project i to represent the start and end time of the project, respectively. Each a_ij has a deterministic duration d_ij.
• There are s (g = 1, 2, …, s) kinds of global resources shared by m projects. Global resource g is available with constant capacity RR_g. In addition, there are k (l = 1, 2, …, k) types of local resources which only occupied by project i. Each local resource l has a constant capacity of R_il. Every a_ij requires units of local resources l and units of global resources g during its whole duration d_ij. Without loss of generality, the durations and local/global resource requirements of the two dummy activities in each project are zero. The resources involved in this study are renewable resources that are not consumed as the progress of project.
• E_ij refers to the predecessor activity set of a_ij. Only when all the activities in E_ij have been completed can a_ij be started.
• T denotes the whole planning horizon. It is the upper bound of all projects completion time. t (t = 0, 1, …, T) is the time point in the process of scheduling.

In addition, the final solution for the proposed DRCMPSP is represented by the vector S = (S₁, S₂,…, S_m), where S_i = (s_i0, s_i1,…,) denotes the schedule of project i and s_ij is the start time of activity a_ij.

Therefore, the total tardiness cost (TTC) is defined as below to evaluate the sequential game results for global resources coordination. (1) where (-ad_i) denotes the final makespan of project i and CPL_i stands for the critical path length of project i [5,12]. Therefore, tc_i ⋅ (-ad_i-CPL_i) is the tardiness cost incurred as the delay penalty of project i.

Agent-based two-stage decomposition approach

Multi-agent system for distributed decision process

As demonstrated in Fig 1, the MAS consists of two types of self-interested and autonomous agents namely: Project Agent (PA) and Coordinator Agent (CA). If there is competition for shared resources among PAs, they are coordinated by CA to achieve the global objective of multi-project scheduling.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Multi-agent information and communication.

https://doi.org/10.1371/journal.pone.0205445.g001

1. Project Agent (PA): Each PA is an independent local decision maker who plays the role of project manager in the company. There are as many PAs as projects. A PA is in charge of his own local information and responsible for generating a satisfactory initial local schedule privately. In addition, the PAs compete for global resources at each conflicting time slot and then modify their local schedule dynamically according to the coordination message received from CA. All the local information and scheduling scheme of individual PAs are secret to each other.
2. Coordinator Agent (CA): CA is a global decision maker who acts as the senior manager in the company. He holds the public information about the global objective and global resource availabilities. CA is mainly responsible for identifying the conflicting time slots, coordinating PAs to eliminate conflicts, and allocating global resources to related PAs.

In addition, all global resources are stored in the global resources pool. It is an important part in the MAS and opens for CA to offer global resources to each PA.

Global coordination decision-making stage

In the real world, each local initial schedule generated in stage one could be destroyed owing to the limited global resources. Therefore, it is indispensable to design an effective coordination mechanism for shared resources allocation without conflicts. In this section, the sequential game-based negotiation mechanism is proposed to eliminate the global resource conflicts among projects. Specifically, CA as a coordinator organizes sequential games for PAs when a shared resource conflict occurs. After several sequential game negotiations, CA determines the best subgame perfect Nash equilibrium according to the global decision model. Then multiple PAs resolve their local schedules with the allocated global resources from CA independently.

Sequential game-based negotiation mechanism.

In game theory, the sequential game is defined as a game consisting of finite and at least two players where each player takes actions at different times or in turn. In other words, one player selects his action before other players choosing theirs, which means that players who make move later have additional information about the actions of those who make move before it in the game [51]. As a rational player, the toper one in the sequence has first-mover advantages since his global resources requirements can be satisfied firstly. In DRCMPSP, we regard the time as discrete resources, namely time slots of a resource. The conflicting time slot is a decision point where the global resource requirements exceed the available amounts. The sequential game among multiple projects is composed of five key elements.

• Players: At each resource conflicting time slot, the PAs who require the limited global resources to schedule their activities are treated as the sequential game players.
• Strategies: The feasible start time slots of conflicting activities at which no resource conflict occurs are defined as strategies or actions for PAs.
• Sequence: Every player takes action in an order. A player chooses its strategy depending on actions taken by other players ordered before him.
• Payoff: It represents the player’s preferences for chosen strategies. Since each PA aims at completing the project as soon as possible, the opposite of the project tardiness cost is evaluated as the payoff of associated PA.
• Equilibrium: The combined strategies selected sequential rationally by all the game players for their activities at a certain conflicting time point are described as the subgame perfect Nash equilibrium.

For a sequential game, the subgame is defined as a subset of the game that contains an initial node and all its successor nodes. Nash equilibrium describes the players’ behavior in a steady state, where any player has no incentive to change his/her own decision. A subgame perfect equilibrium is an equilibrium strategy profile that in no subgame can any player do better by taking an action different from the profile while every other player adheres to it [51], according to which the resource conflict is resolved. To obtain the subgame perfect equilibrium for a sequential game, the Nash equilibrium is defined for each subgame as below.

Definition (Nash equilibrium). Let A be an action profile including all feasible start time slots of conflicting activities for game players, in which the action set of each player PA_i is A_i. Let and aa_i (∀aa_i ∈ A_i) be the optimal and any feasible start time slots chosen by player PA_i, respectively. Then () represents the action profile in which all players other than PA_i adhere to choose the best action of (∀q ∈ {1,…,i − 1,i + 1,…,m}) while PA_i chooses aa_i. If , then of course () = () = aa* (). In a normal-form game, the action profile aa* is a Nash equilibrium if, for each player PA_i and any other feasible time slots actions aa_i ∈ A_i, is the best response of player PA_i to the strategies specified for the other players. Equivalently, for each player PA_i and every feasible time slots action aa_i of player PA_i, (9) where u_i is a payoff of player PA_i.

In the light of the sequential structure, each player’s strategy need to be optimal in the sequential game to avoid a perturbed steady state or achieve the Nash equilibrium. The optimal strategy chosen by each PA is proposed and proved as follows.

Proposition. In a sequential game, the strategy best for any given player depends on the other players’ actions. The subgame perfect Nash equilibrium is determined if, every player is sequentially rational which means that no matter what happened in the past, each player PA_i prefers to choose the optimal strategy i.e. the earliest available time slots according to the decision-making sequence. At each conflicting time slot, the PAs involved in the sequential game will employ the virtual scheduling method (see Section “Resource conflicts dynamic elimination algorithm”) to resolve their respective local schedules.

Proof. Suppose that it’s time for PA_i to make decision for activity a_ij, PA_i chooses a strategy other than the best one which is later than the earliest available time slots. Then a_ij is scheduled at the time slot determined by the chosen strategy. According to the virtual scheduling method, unscheduled activities of PA_i belonged to the successors of a_ij will not start earlier than its local initial start time obtained in stage one, which means the project makespan cannot be shortened. According to Eq (1), the tardiness cost of project i will not be reduced. In other words, no additional payoffs are gained for PA_i. As a rational project agent, no changes will be made to realize his local objective and a steady state is achieved. Therefore, in each iteration of sequential game negotiation, the strategy combination that every player chooses the earliest available time slots orderly is consist of subgame perfect Nash equilibrium.

Global decision model.

For each conflicting time slot, CA, acts as a senior manager, is responsible for finding a best sequence with the best subgame perfect Nash equilibrium to meet the following global decision model: (10) where objective function (10) seeks to minimize the TTC of the multi-project scheduling S_v (π, t_c) obtained by using a virtual scheduling method (see Section “Resource conflicts dynamic elimination algorithm”) at the conflicting time slot t_c following a sequence π ∈ Π. It is worth noting that the global resource feasibility at t_c is guaranteed by the virtual scheduling method.

Resource conflicts dynamic elimination algorithm.

After all the PAs generating their local initial schedules in stage one, a local activity list of each PA is generated by ordering activities with ascending start times. As shown in Fig 2, the information of global resource requirements of PAs is submitted to CA. Afterwards, CA identifies the resources conflicts and eliminates them one by one. For a sequential game involving N players, there will be a total N! of possible sequences of players, which is well-known to be NP-hard. Therefore, a randomized search heuristic method is used to determine the best sequence and best subgame perfect Nash equilibrium. More concretely, at the first conflicting time slot, CA allocates global resources to the corresponding projects by initiating multiple iterations of sequential game-based negotiation. In each iteration, the sequence of game players is determined randomly. Then the projects involved in the sequential game are virtually scheduled by a given method. Specifically, for each PA, activities starting earlier than the conflicting time are virtually scheduled as its local initial schedule, activities starting at the conflicting time are virtually scheduled according to the subgame perfect Nash equilibrium, and activities starting later than the conflicting time are virtually scheduled by a fast heuristic procedure with SSGS based on the local activity list determined in stage one. After several iterations of negotiation, CA determines the best subgame perfect Nash equilibrium with minimal TTC according to the global decision model to eliminate the current resource conflict. Followed by CA allocating global resources, each of the corresponding PA modifies its local schedule by the above method. The following conflicts are identified and solved in the same way. Until there is no conflicts at all time slots, the whole process is terminated. Details of the conflicts dynamic elimination process are described as Algorithm 2.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Flowchart of conflicts elimination.

https://doi.org/10.1371/journal.pone.0205445.g002

Algorithm 2: Resources conflicts dynamic elimination.

Initialize: current time t ← 0.

Step 1. Identify conflicting time slots.

    while t≤T do

        Each PA submits its global resource requirements information to CA;

        Determine the earliest conflicting time slot t_c;

        Go to step 2;

    end while

Step 2. Perform sequential game-based negotiation on PAs.

(a) Determine the set of game players GP.

(b) Initialize negotiation iteration N ← 0, Max iterations←Ω.

(c) Virtually schedule each local project.

    while N≤Ω do

        Determine a game sequence;

        Solve subgame perfect Nash equilibrium;

    for each PA_i in GP do

        Activities with start time s_ij<t_c are virtually scheduled as its local initial schedule;

        Activities with start time s_ij = t_c are virtually scheduled according to the subgame perfect Nash equilibrium;

        Activities with start time s_ij>t_c are virtually scheduled by SSGS based on the local initial activity list;

    end for

        Compute TTC_N according to Eq ();

        N ←N +1;

    end while

Step 3. Coordinate global resources allocation.

(a) Determine the best subgame perfect Nash equilibrium with minimal TTC.

(b) CA allocates the global resources to corresponding PAs.

(c) Each PA modifies and updates its local initial scheduling.

(d) t ←t +1;

    Return to step 1.

Output: final multi-project schedule and final TTC.

An illustrative example

In this section, we present a simple example to demonstrate the proposed approach for DRCMPSP. In addition, we use this example to illustrate the fact that the multi-project schedule that minimizes APD cannot guarantee a minimum TTC for the senior manager. An illustrative multi-project scheduling problem with two projects is considered. Each project consists of five activities where two of them are dummies. Each activity requires units of local resources and units of global resources. Fig 3 demonstrates the activity-on-node (AON) networks and all information of the two projects. The duration d_ij and requirements of local/global resources and of each activity are marked on the top and bottom of the corresponding node, respectively. Dummy activities have zero duration and resources requirements. Other information about arrival date ad_i, unit tardiness cost tc_i, and local resource capacities R_il of each project is presented in the box left to the corresponding project AON network. As shown between the two boxes, the capacity RR₁ of global resource is assumed to be 15.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Information for illustrative example.

https://doi.org/10.1371/journal.pone.0205445.g003

In the following, we elaborate the whole two-stage decision-making process.

1. Stage one: Local initial decision-making process

As shown in Fig 4, each PA generates an initial schedule independently to minimize the project completion time. It can be seen that the makespan of project 1 and project 2 are 10 and 6 respectively, which are equal to the critical path length (CPL_i) of each corresponding project. Obviously, the global resource requirements violate the given capacity from time slot 6. Therefore, it is necessary for CA to eliminate the global resource conflicts.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Initial schedule of each project.

https://doi.org/10.1371/journal.pone.0205445.g004

1. Stage two: Global coordination decision-making process

After each PA produces its local initial schedule without consideration of sharing global resources with other PAs, CA receives the global resource requirements information from each PA. The first conflicting time slot 6 is identified firstly by CA, and then the process of sequential game-based negotiation between PA₁ and PA₂ begins. At the first conflicting time slot 6, game players PA₁ and PA₂ decide when to start a₁₃ and a₂₁, respectively. Time slots after time point 5 constitute the strategies for each PA to execute its corresponding activities. Suppose the game sequence is {PA₁, PA₂} in the first iteration, the subgame perfect Nash equilibrium is determined, i.e. a₁₃ and a₂₁ are started at time slot 6 and time slot 11 by PA₁ and PA₂, respectively. It can be calculated that the TTC of the virtual multi-project schedule in the first iteration is 50. In the same way, if the game sequence is {PA₂, PA₁} in the second iteration, the subgame perfect Nash equilibrium is determined, i.e. PA₁ schedules a₁₃ at time slot 8 and PA₂ schedules a₂₁ at time slot 6. The TTC of virtual multi-project schedule in the second iteration is 10. The virtual multi-project schedule in each iteration is shown in Fig 5. Then CA determines the best subgame perfect Nash equilibrium with the objective of minimizing TTC. Clearly, CA prefers to allocate global resources according to the negotiation result in the second iteration (Fig 5(B)). Similarly, the virtual schedule in each iteration after eliminating the second conflict is demonstrated in Fig 6. CA will choose the schedule in Fig 6(B) as its final decision since the TTC value is smaller.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Virtual schedule of multi-project after eliminating the first conflict at time slot 6.

https://doi.org/10.1371/journal.pone.0205445.g005

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Virtual schedule of multi-project after eliminating the second conflict at time slot 8.

https://doi.org/10.1371/journal.pone.0205445.g006

Since the unit delay penalty may differ by projects, a multi-project schedule with minimized APD is not aligned with the objective of TTC minimization. As illustrated in Fig 5(A), the APD in this case is minimized with the value of 2.5 while the TTC is 50. In Fig 6(B), the schedule is determined with the minimum TTC of 30 while the APD is 3. Therefore, our study focus on minimizing the TTC objective of multi-project from the portfolio perspective rather than devoting to obtain lower APD on the project level.

Computational study

In order to evaluate the performance of the proposed approach, we carry out a series of computational experiments. Algorithms are coded in Matlab R2013a. All the experiments are executed on a single computer with Inter Core i7 (3.4GHz, 8GB RAM). The performance of the solutions is evaluated in two aspects:

1. The local initial schedule obtained in stage one are evaluated according to project makespan (MS_i). The results are also compared with the optimal solutions obtained by the B&B algorithm to demonstrate the quality of the schedules obtained by using FBHGA.
2. The performance of the proposed sequential game-based negotiation mechanism is evaluated based on the measure of TTC. We conduct the pre-experiment by using different numbers of negotiation iterations. We also compare the performance of our algorithm in terms of TTC with two state-of-the-art variants of the coordination mechanism integrated in a new multi-agent system (CMAS) based decentralized methods: CMAS/ES with the (μ,λ)-coordination mechanism [18], and CMAS/SA with the mechanism proposed by Fink [52]. Since these two existing methods focus on minimizing APD, we use the feasible multi-project schedules produced by the two methods which are available from MPSPLIB (http://www.mpsplib.com, last check of address: 16 November 2017) to compute the TTC of each multi-project instance according to Eq (1) for comparison.

Problem instances

The proposed algorithm is tested on all the 140 problem instances from MPSPLIB (Supporting Information file S1 Appendix). As shown in Table 1, these instances are classified into 20 subsets according to the Utilization Factors (UF). Each problem subset is named as “MPJ_i_m” (MP subsets) or “MPJ_i_mAC” (MPAC subsets), where the number of activities J_i per project is 30, 90, or 120, and the number of projects m is 2, 5, 10, or 20. Each problem instance is provided with s and k types of global and local resources, respectively. UF_g is the utilization factor of global resource g∈G indicating the ratio of the total requirements rr^g to the constant capacity RR_g available in each time period over the global critical path [18]. UF is calculated as Eq (11) denoting the maximum tightness of the constraints on required global resources [53].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Problem instances for DRCMPSP (Homberger 2012).

https://doi.org/10.1371/journal.pone.0205445.t001

(11)

The UF_AV in Table 1 is the average of the UF values of a given problem subset. UF >1 indicates a medium to high overload which means the global resource conflicts among multiple projects are relatively strong, while UF <1 denotes a low to medium overload which means the conflicts are relatively weak [12].

Results and analysis

The performance of the proposed two-stage scheduling algorithm for DRCMPSP is evaluated on two stages separately. For algorithm 1, the best parameter settings of GA are determined by pre-experiments as follows: the population size POP₀ = 60, the number of generations Gen = 100, the crossover and mutation rates are 0.9 and 0.1 respectively. For algorithm 2, to obtain the final solution in reasonable CPU running time, we set the number of negotiation iterations Ω to three levels: 1, 5, and 10, respectively.

Comparison of local initial schedule.

To compare the results in terms of project makespan of each local initial schedule obtained in stage one, we have performed the exact B&B algorithm and the proposed FBHGA on 1295 single projects from the 140 multi-project problem instances. The optimal solutions of some small-size single project instances are obtained by the B&B algorithm embedded in RESCON software [19]. Note that the solvable solutions obtained by RESCON are all optimal results and any solutions cannot be found if the RESCON runs out of memory.

Table 2 shows the comparison of the average project makespan (APM), the number of projects with optimal solutions (NPOS), and the number of projects with no feasible solutions (NPNS) solved by B&B and FBHGA respectively, meanwhile the last column shows the average relative deviation (ARD) of the non-optimal solutions obtained by FBHGA from the optimal results reached by B&B of each subset. It is clear that out of the 1295 single projects, B&B can obtain optimal solutions for 954 projects and 89.3% of which are also obtained by FBHGA. In addition, the more the activities involved in a project, the less the possibility that a feasible solution can be obtained by B&B before running out of memory, especially for projects with 120 activities. However, the FBHGA can achieve a feasible scheduling solution efficiently for any size project case. For each problem subset, the results of APM obtained by FBHGA and B&B are very close with respect to the solvable solutions. Furthermore, the ARD of each subset is lower than 5%. The average of ARD over all of the 20 subsets is less than 1% (0.81%). According to the statistical analysis of T test, there is no significant difference between B&B and FBHGA solutions (the P_value is 0.750). It fully turned out that the FBHGA performs well on generating high-quality local initial schedules with satisfactory makespan for every PA in stage one.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 2. Comparison of results obtained by B&B and FBHGA in stage one.

https://doi.org/10.1371/journal.pone.0205445.t002

Performance analysis of coordination mechanism.

It should be noted that the number of iterations Ω for sequential game negotiation is considered in three modes i.e. 1, 5, and 10. As revealed in Table 3, with the increasing of iterations, the average TTC of each subset reduces while the CPU running time increases. The effect of the number of iterations on either TTC or CPU running time is statistically significant based on the related-samples Kendall’s coefficient of concordance test (the P_value of each test is 0.000). The reason is that the more the negotiation iterations, the greater the possibility of finding better solutions and thus the more CPU time is consumed. Compared to the results with Ω = 1, the improvements on TTC of MP subsets and MPAC subsets with Ω = 5 and Ω = 10 are depicted respectively in Fig 7. Clearly, the TTC results with Ω = 10 are improved more than that with Ω = 5 by comparing the corresponding results with Ω = 1 for all subsets. Additionally, the greater improvements with both Ω = 5 and Ω = 10 are made for large-size subsets with high UF_AV. Fig 8 illustrates the CPU time results of MP subsets and MPAC subsets with different sets of iterations. It is clear that the larger the size of subsets, the longer the CPU running time is. Meanwhile, the MPAC subsets generally run longer than the MP subsets with the fact that more conflicts need to be solved for MPAC subsets as a result of the higher UF_AV. Since the better solution of each multi-project instance can be obtained within about 20 min, we set Ω = 10 for following study analysis.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Comparison of TTC improvements of MP subsets and MPAC subsets.

https://doi.org/10.1371/journal.pone.0205445.g007

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Comparison of CPU time of MP subsets and MPAC subsets.

https://doi.org/10.1371/journal.pone.0205445.g008

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Computational results with different number of negotiation iterations.

https://doi.org/10.1371/journal.pone.0205445.t003

Given the complexity and diversity of the practical multi-project scheduling environment, senior managers in the company often have to deal with multiple projects under different degrees of conflicts [54]. To evaluate the performance of the proposed sequential game-based negotiation mechanism on different instance subsets with various problem sizes and degrees of conflicts, we analyze two indicators: DTTC and NC. DTTC is the deviation of the final actual total tardiness cost (TTC(S), obtained in stage two) from the initial total tardiness cost (TTC(S_initial), obtained in stage one). NC represents the number of conflicts occurred in the whole multi-project scheduling horizon.

According to the values of UF_AV, the 20 problem subsets are classified into two groups: subset group with UF_AV <1 and the other one with UF_AV >1. As shown in Fig 9, the average values of DTTC and NC of the subset group with UF_AV <1 are significantly lower than those values of the subset groups with UF_AV >1. It reveals that the stronger the degrees of conflicts among multiple projects, the greater the final actual TTC value deviates from the initial TTC value, and the more the number of conflicts existing in the process of multi-project scheduling. The results indicate that the multi-project scheduling environment will become complicated and more TTC will be incurred when the global resource conflicts among projects are relatively strong.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Comparisons of DTTC and NC under different degrees of conflict.

https://doi.org/10.1371/journal.pone.0205445.g009

Fig 10 illustrates the relationship between problem size and TTC result for all subsets. According to the one-way ANOVA test, the problem size has no significant effect on TTC (the P_value is 0.558), which is mainly because the unit tardiness cost of each project in a multi-project instance is different.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 10. Relationship between problem size and TTC result.

https://doi.org/10.1371/journal.pone.0205445.g010

In order to evaluate the performance of the proposed two-stage decomposition approach with respect to TTC, the comparisons with two existing decentralized methods i.e. CMAS/ES and CMAS/SA are shown in Table 4. The best results of each subset are marked in bold, those of strong conflicts (UF_AV >1) and large-size multi-project instances are shown with * sign and underline, respectively. As can be seen in Table 4, for 15 out of 20 subsets, the proposed approach performs best. Specifically, for 12 out of 14 subsets with strong conflicts and for 7 out of 8 subsets with large-size multi-project instances, the proposed approach obtains a much lower TTC than any other methods. To clearly show the better performance with regard to the instance size and resource utilization factor, we compute the percentage improvement of our method comparing with CMAS/ES and CMAS/SA, respectively. The statistical comparisons for small- and large-size instances, and for UF_AV <1 and UF_AV >1 instances are depicted in Fig 11 and Fig 12, respectively. In each figure, the maximum and the average of percentage improvements are demonstrated on the left (a) and right (b), respectively. It is indicated that for both CMAS/ES and CMAS/SA, our method shows superior performance on large-size and strong conflicts instances. Moreover, for all subsets, the percentage improvements of our approach in reducing TTC achieve an average of 10%-15% compared to CMAS/ES, and a maximal of 20%-36% compared to CMAS/SA. From above results, the two-stage decomposition approach outperforms other decentralized methods in providing satisfactory solutions in terms of TTC, especially for large-size and strong conflicts instances.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 11. Percentage improvement with respect to different problem sizes.

https://doi.org/10.1371/journal.pone.0205445.g011

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 12. Percentage improvement with respect to different degrees of conflicts.

https://doi.org/10.1371/journal.pone.0205445.g012

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 4. Comparisons of TTC among different decentralized methods.

https://doi.org/10.1371/journal.pone.0205445.t004

Conclusion and future work

This paper is devoted to solving DRCMPSP using a two-stage decomposition approach. Individual project managers focus on minimizing the project completion time initially while the senior manager pays more attention to reducing TTC from a global perspective. A FBHGA is introduced to determine the local initial schedule for each PA in stage one, and a sequential game-based negotiation mechanism is proposed to coordinate the global resources allocation among multiple projects in stage two.

The performance of the proposed approach has been evaluated on the MPSPLIB benchmark. High-quality local initial schedules can be generated by FBHGA for every individual PA and the average ARD with respect to the optimal solutions over all problem subsets is less than 1%. By conducting preliminary experiments on all subsets, satisfactory solutions in terms of TTC can be generated in reasonable CPU running time after 10 iterations of sequential game-based negotiation. The senior manager will face complex multi-project management situation and more TTC will be incurred when the shared global resources are relatively scarce. The performance of our approach is compared with two existing decentralized methods with different coordination mechanisms. Experiments reveal that the proposed approach with sequential game-based negotiation mechanism can significantly reduce TTC, especially for the instances with large problem size and high resource utilization factor.

As the increasing of multinational corporations in the process of globalization, more and more multiple projects are executed simultaneously at different locations. Our proposed two-stage decomposition approach provides decision guidance for distributed multi-project scheduling managers. Meanwhile, the sequential game-based negotiation mechanism opens a new perspective for shared resource allocation. However, there is a limitation that the randomized search heuristic procedure is used to obtain a best subgame perfect Nash equilibrium result. In the future, we aim to find the approximate optimal solutions by efficient meta-heuristics. In addition, considering the possible uncertain environments in today’s globally active industries, the proposed two-stage decomposition approach with sequential game-based negotiation mechanism opens up new directions for stochastic distributed multi-project scheduling problem.

Supporting information

Acknowledgments

We would like to thank our team for excellent technical support and assisting in preparation of this manuscript.