Multi-Criteria Decision Support for Fairness-Aware Coordination in Distributed Resource-Constrained Multi-Project Scheduling

Abstract

In engineering R&D organizations, resource contention among autonomous projects creates complex scheduling challenges in distributed multi-project environments. Traditional coordination mechanisms often prioritize global efficiency while overlooking inter-project fairness, which can lead to stakeholder resistance and execution delays. This study proposes a transparent decision support approach for the Distributed Resource-Constrained Multi-Project Scheduling Problem (DRCMPSP). We develop a two-stage scheduling mechanism that integrates Multi-Criteria Decision-Making (MCDM) into the global coordination process. First, an enhanced Chaotic Genetic Algorithm (CGA) with elitism generates local schedules. Second, global resource conflicts are resolved using MCDM methods. Inter-project fairness is implemented by limiting each project’s relative objective deterioration and by evaluating the dispersion of the resulting project-level burdens. Validation through an Unmanned Aerial Vehicle R&D case study and extensive experiments shows that the TOPSIS-based mechanism achieves a significantly lower standard deviation than the auction-based mechanism under high resource contention. The approach supports transparent and fairness-aware conflict resolution by making trade-offs among project-level outcomes explicit to managers.

1. Introduction

In engineering R&D organizations, multiple projects are often executed in parallel. Some firms centrally rent or reserve access to external public testing or pilot-scale experimental platforms for verification and prototype validation, and then allocate the limited service time among parallel projects. Because platform capacity depends on certified personnel, high-end equipment, and available testing slots, competition for limited service windows creates a practical need to coordinate scarce resources transparently while preserving schedule efficiency and project autonomy. This allocation challenge is modeled as the Resource-Constrained Multi-Project Scheduling Problem (RCMPSP). Traditional centralized approaches assume that a single decision-maker has full authority and information to optimize global objectives. However, such approaches struggle to balance conflicting performance metrics across interdependent projects and frequently overlook the distinct priorities of individual projects regarding cost, time, and quality [1,2]. Therefore, global schedules that disregard local interests may lack stakeholder acceptance. The Distributed RCMPSP (DRCMPSP) addresses this limitation by modeling the overall system as a set of autonomous agents, each with distinct objectives.

Fairness is a critical consideration in resource-allocation and scheduling decisions involving scarce capacity and competing stakeholders [3,4]. In distributed multi-project systems, uneven project-level outcomes after shared-resource coordination can weaken stakeholder acceptance and coordinated execution. In collaborative engineering, scheduling involves interactive decision-making [5], during which perceptions of fairness often arise from social comparisons. Inequity Aversion Theory suggests that individuals exhibit a strong aversion to unfair distributions and may incur costs to reject them [6,7]. When a schedule forces sub-projects to sacrifice critical milestones for the collective benefit without justification, it undermines the psychological contract among stakeholders. Such perceived inequity may induce dissatisfaction and trigger counterproductive behaviors, such as passive resistance or unauthorized schedule modifications, thereby compromising coordinated multi-project execution. Transparent coordination is important because affected project agents should be able to understand the basis of shared-resource allocation decisions, thereby supporting informed use of the scheduling recommendation [8,9].

New energy equipment manufacturing provides a concrete example of this coordination problem. A firm may reserve testing time on a shared public platform and allocate it among parallel teams developing energy storage prototypes. Because the platform offers limited windows for environmental and reliability verification, earlier slots may be assigned to teams with stronger bargaining power rather than to schedule-critical tasks. Disadvantaged teams may perceive this allocation as unfair and become less committed to the coordinated schedule. Attempts to recover lost time through experience-based technical adjustments may create additional losses and disrupt the original multi-project schedule. This demonstrates how perceived unfairness in resource allocation can destabilize the entire project system. Therefore, a scheduling system that accounts for heterogeneous project objectives is essential to both fairness and orderly project delivery.

Multi-Agent Systems (MAS) are used to model the DRCMPSP because they reflect the distributed knowledge and decision authority in engineering R&D organizations [10,11]. The problem is typically decomposed into two stages: local scheduling and global coordination. Project Agents (PAs) schedule the activities of their respective projects, while a Coordination Agent (CA) mediates global resource conflicts and enforces aggregate shared-resource capacity. The key challenge is to design a coordination mechanism that resolves conflicts efficiently while explicitly considering fairness across project outcomes and making the coordination basis transparent to PAs.

Most DRCMPSP studies use auction-based or negotiation-based mechanisms to resolve global resource conflicts [12]. Auction-based mechanisms model global resource time slots as bid items [13]. While computationally efficient, they often favor projects with larger budgets or specific cost structures. This tendency can systematically disadvantage smaller, schedule-critical projects. Negotiation-based mechanisms rely on iterative bargaining under a specified protocol to determine resource allocations [14]. Typical implementations include critical chain activity scoring [15], general scoring mechanisms [16], and game-theoretic approaches [17].

A gap remains in coordination mechanisms that combine transparent ranking of feasible alternatives with explicit consideration of fairness across heterogeneous project objectives. In distributed settings with information asymmetry among autonomous project agents, decisions for resolving resource conflicts should be transparent and justifiable. Multi-criteria decision-making (MCDM) offers a structured framework for evaluating trade-offs among competing objectives, making it well-suited for systemic conflict resolution. In this context, MCDM supports transparent conflict resolution, and the standard deviation provides a post-coordination assessment of outcome balance. However, the application of MCDM to coordination in DRCMPSP remains limited.

To address this research gap, this study proposes a two-stage scheduling approach for the DRCMPSP. First, we design a fairness-aware global coordination mechanism using MCDM. This mechanism employs the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) to evaluate competing resource requests objectively. The standard deviation is used to evaluate the balance of project-level outcomes. Second, to enable frequent rescheduling, this study develops an enhanced CGA with an adaptive permutation-mutation mechanism for efficient local scheduling. The mutation probability adapts to evolutionary progress and population fitness variance, while Logistic-map values select among swap, insertion, and inversion operators. Numerical experiments and a case study show that the TOPSIS-based mechanism achieves a significantly lower standard deviation than the auction-based mechanism under high resource contention. This study provides a structured and transparent decision-support approach for coordinating distributed multi-project schedules.

This research is organized as follows. Section 2 reviews the relevant literature. Section 3 presents the problem description and the mathematical formulation. Section 4 details the proposed MCDM-based two-stage scheduling approach. Section 5 illustrates the application of the method through a case study on the R&D of a UAV series. Section 6 reports the results of the numerical experiments. Finally, Section 7 presents conclusions and outlines directions for future research.

2. Literature Review

2.1. Objective Functions and Priority Analysis in RCMPSP and DRCMPSP

Objective functions and priorities reflect stakeholder demands and significantly influence scheduling fairness [18,19]. Unlike the single-project scheduling problem, RCMPSP is characterized by objective heterogeneity and inter-project conflicts [20]. Objectives in RCMPSP are generally categorized into time-based, economic, and project-related dimensions. Time-related objectives typically include minimizing the multi-project makespan, average delay percentage, and deviation from expected duration, and maximizing the net present value [21,22]. Makespan and total completion time are widely used metrics and are often regarded as proxies for customer satisfaction and resource utilization. However, simultaneously minimizing both metrics typically requires rather stringent conditions [23,24]. Projects in RCMPSP are prioritized and ranked according to coordinators’ objectives and the project environment. Accordingly, these objectives are reflected in the priority rules and evaluation metrics.

Heuristics and metaheuristics are often applied to large-scale scheduling settings or problems with conflicting objectives [25], where priority rules play a central role. In DRCMPSP, the local scheduling of individual projects follows the classic RCMPSP, with metaheuristic algorithms such as particle swarm optimization and genetic algorithms commonly used to obtain high-quality feasible solutions efficiently, often through problem-specific adaptations [26,27]. Kolisch [28] proposed priority-based heuristics and assessed the performance of various priority rules. Subsequent studies have broadened the range of priority rules, showing that the best-performing rule depends on specific project characteristics [29,30]. However, when global multi-project objectives are pursued too aggressively under a single-priority rule, resources may be preferentially allocated to projects with many activities or substantial delays. From the standpoint of stakeholder management, such prioritization schemes may nevertheless be perceived as unfair.

Fairness-oriented operations research considers the distributional consequences of allocation decisions together with system efficiency. Price-of-fairness analysis quantifies the efficiency loss associated with fairness requirements. Resource-allocation studies have examined proportional and max-min fairness, while a recent two-agent scheduling study considers Kalai–Smorodinsky and proportional fairness when two agents compete for a single machine [31,32]. Equity concerns can enter an optimization model through worst-off outcomes, inequality indices, or aggregation functions that balance equity and efficiency [3]. The choice of a fairness formulation depends on the distributional and axiomatic properties relevant to the decision setting [33]. In capacity-constrained scheduling, bi-objective models jointly evaluate scheduling efficiency and fairness at aggregate and individual-user levels [4]. These studies provide context for incorporating project-level rescheduling losses into DRCMPSP coordination.

Fairness metrics emphasize different distributional properties. Max-min criteria protect the worst-off project. The Gini coefficient measures inequality in the outcome distribution. Jain’s index is a monotone transform of the coefficient of variation and increases as outcomes become more equal [33]. Standard deviation describes the dispersion of project-level outcome values around the within-instance mean and gives greater influence to larger departures. This property makes it informative when the coordination concern is whether rescheduling losses are concentrated on a few PAs. Dauzere-Péres et al. [34] identify variance around average machine workload as a fairness measure in flexible job shop scheduling. Accordingly, this study adopts standard deviation as the balance indicator for project-level outcomes in DRCMPSP coordination.

2.2. Coordination and Decision Mechanisms in DRCMPSP

Global scheduling mechanisms in DRCMPSP, referred to as coordination mechanisms, are classified into auction-based and negotiation-based approaches. Auction-based mechanisms allocate global resources through price competition under established rules, such as combinatorial auctions [13,35]. Negotiation-based mechanisms iteratively negotiate allocation schemes for conflicting resources following specific protocols [14]. Accordingly, negotiation-based mechanisms offer greater extensibility and flexibility than auction-based approaches in addressing the diverse requirements of decision-makers.

Some negotiation-based mechanisms incorporate decision-makers’ preferences, thereby enhancing fairness. Homberger and Fink [36] designed an automated negotiation mechanism between two project agents that accounts for agent preferences and conflicts of interest. Li et al. [14] proposed a multi-agent coordination mechanism based on negotiation protocols to mitigate selfish behaviors associated with single-agent global resource coordination. Game-theoretic coordination models strategic incentives and equilibrium outcomes through specified payoff functions and interaction protocols [17,37].

Multi-criteria trade-offs have also been studied in single-project scheduling. Monghasemi et al. [38] generated Pareto solutions with a multi-objective genetic algorithm and ranked them using evidential reasoning, whereas Kebriyaii et al. [39] compared multi-objective metaheuristics. These studies evaluate time-cost-quality trade-offs among final alternative schedules.

Table 1. Overview of related literature on DRCMPSP.

Global Scheduling Mechanism	Objectives	Local Scheduling Method	Problem Instances	Literature
An iterative negotiation mechanism based on tentative contract	Minimizing the discounted cash flow	A genetic algorithm	PSPLib	Homberger and Fink [36]
A simple sequence learning game	Minimizing the average project delay and the total makespan	A reinforcement learning algorithm	MPSPLIB	Wauters et al. [40]
A sequential game-based negotiation mechanism	Minimizing total tardiness cost	A forward-backward hybrid genetic algorithm	MPSPLIB	Li and Xu [17]
A repeated-negotiation non-cooperative game	Minimizing makespan and schedule total cost	Agent-based simulation	A case study	Tosselli et al. [37]
A priority-based task-scoring mechanism	Minimizing the average project delay	A genetic algorithm with forward and backward scheduling	MPSPLIB	Liu et al. [12]
A Markov decision process modeling	Minimizing the total makespan	An up-to-date deep reinforcement learning algorithm	Eighteen instances	Wang et al. [41]
A multi-criteria decision-making method	Minimizing distinct progress and cost objectives	A chaotic genetic algorithm	MPSPLIB	This study.

summarizes related DRCMPSP coordination mechanisms. It focuses on studies that explicitly coordinate multiple projects under shared resources, while the single-project studies provide related MCDM and multi-objective scheduling context.

2.3. Summary

To sum up, existing DRCMPSP studies provide auction, negotiation, game-theoretic, and preference-scoring mechanisms, but offer limited support for transparently ranking several complete schedules against heterogeneous project objectives. MCDM addresses this specific gap by ranking feasible alternatives, while outcome balance is assessed after coordination. This study introduces a two-stage scheduling approach that incorporates MCDM methods. Specifically, we use the available decision information to compare objective losses across projects in DRCMPSP.

3. Model Formulation

3.1. Problem Settings and Assumptions

To investigate perceived unfairness arising from objective losses, we consider a DRCMPSP with heterogeneous project objectives. In this setting, projects are scheduled autonomously, and conflicts over shared resources are resolved through coordination. Accordingly, we model the problem using a multi-agent system (MAS) framework. Specifically, each Project Agent (PA) formulates a resource-usage plan and submits it to the Coordination Agent (CA). Each PA enforces local-resource feasibility. For rescheduling, the CA sends the remaining period-specific global-resource availability after deducting the demand of projects whose schedules have been fixed. The CA evaluates the aggregate demand for global resources and coordinates projects when their combined demand exceeds capacity. After receiving the updated resource availability, PAs reschedule and report the revised resource consumption. This iterative process continues until the total demand for global resources does not exceed the supply limit, yielding a feasible multi-project schedule. Figure 1 illustrates this conceptual framework, which comprises the information exchange mechanism, the local scheduling model, and the global scheduling model based on multi-criteria decision-making (MCDM).

In engineering R&D settings, the term “distributed” denotes decentralized project management rather than spatial dispersion. Global resources in R&D portfolios, such as specialized personnel and equipment, may be physically centralized but are shared by autonomous projects. Consequently, this study adopts the following assumptions:

1.

Projects interact only through shared global resources. Precedence relations are defined within each project.

2.

All projects and their arrival times are known when the coordination process begins.

3.

Local schedules are formulated independently by each PA, whereas the CA enforces aggregate global-resource feasibility.

4.

Resource transfer times and costs between projects are negligible.

5.

Both global and local resources are renewable; their full capacities become available at the start of each period.

6.

Activities are non-preemptive, which means they cannot be interrupted once initiated.

7.

Each activity has a fixed duration and execution mode, and its period-specific resource requirements are known.

Table 2. Notation and definitions.

Notation	Description
Sets and Indices
$\mathrm{MP}$	Set of projects.
i	Index of project, $i\in \mathrm{MP}$.
${\mathrm{V}}_i$	Set of activities in project i, ${\mathrm{V}}_i=\{0,1,\dots ,J_i,J_i+1\}$.
$J_i$	Number of non-dummy activities in project i.
$A_{i0},A_{i,J_i+1}$	Zero-duration dummy start and dummy end activities of project i, respectively.
j	Index of activity in project i, $j\in {\mathrm{V}}_i$.
t	Index of time period (day), $t=1,\dots ,T$.
${\mathrm{L}}_i$	Set of local resources of project i.
l	Index of local resource, $l\in {\mathrm{L}}_i$.
$\mathrm{G}$	Set of global resources shared by the projects.
g	Index of global resource, $g\in \mathrm{G}$.
Parameters
$A_{ij}$	Activity j of project i.
$d_{ij}$	Duration of activity $A_{ij}$.
$P_{ij}$	Set of immediate predecessors of activity $A_{ij}$.
$A{T}_i$	Arrival time of project i.
$D{D}_i$	Due-date allowance measured from project arrival; its absolute due date is $A{T}_i+D{D}_i$.
$D_i$	Absolute due date of project i, $D_i=A{T}_i+D{D}_i$.
$C_i^{\mathrm{Delay}}$	Unit delay cost of project i.
$D{C}_i^{\mathrm{ref}}$	Positive project-specific reference total delay cost.
$OB{J}_i^0,P{D}_i^0$	Objective value and duration of project i in its initial independent schedule.
${\rho }^{max}$	Maximum allowable relative deterioration in a project objective.
$R_{ilt}^L$	Availability of local resource l for project i at time t.
$R_{gt}^G$	Complete availability of global resource g at time t.
${\overline{R}}_{gt}^G$	Remaining availability after deducting the demand of fixed projects.
$r_{ijlt}^L$	Demand for local resource l by activity $A_{ij}$.
$r_{ijgt}^G$	Demand for global resource g by activity $A_{ij}$.
$\mathbb{I}(·)$	Indicator function, equal to 1 when its condition holds and 0 otherwise.
Decision Variables & Outcomes
$s_{ij}$	Integer start time of activity $A_{ij}$.
$f_{ij}$	Finish time derived as $s_{ij}+d_{ij}$.
$F{T}_i$	Finish time of project i.
$P{D}_i$	Duration of project i.
$PT{D}_i$	Completion-time deviation of project i.
$D{T}_i$	Tardiness (delay time) of project i.
$D{C}_i$	Total delay cost of project i.
Priority Rules & Evaluation
$\mathrm{PR}$	Set of priority scheduling rules.
m	Index of priority rule, $m\in \mathrm{PR}$.
$S_m$	Multi-project schedule generated by rule m.
${\delta }_{im}$	Relative objective deterioration of project i under schedule $S_m$.
$ATR$	Set of attributes for schedule evaluation.
n	Index of attribute.
$f_n$	Number of PAs adopting the objective corresponding to attribute n.
${\omega }_n$	Weight of attribute n.
W	Set of attribute weights.
$\mathrm{EV}(·)$	Multi-criteria evaluation function.

lists the model parameters and notations. The project portfolio in the following text refers to the set of projects as a whole, distinguishing the multi-project entity from individual projects.

3.2. Local Scheduling Model

In the local scheduling stage, each PA formulates an optimization model to determine activity start times as its scheduling decision variables based on its objective. This model enforces precedence relations and the availability of local resources. For the initial schedule, the PA uses the complete global-resource availability $R_{gt}^G$; during coordination, it uses the remaining availability ${\overline{R}}_{gt}^G$ supplied by the CA. The solution yields a project schedule comprising the start times for all activities. Since daily resource demands are exogenous parameters, the resource usage plan is derived directly from the project schedule. For project i ($\forall i\in \mathrm{MP}$), the local scheduling model is formulated as follows.

$$\begin{array}{cc}\hfill & min{OBJ}_i\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& s_{ij}+d_{ij}\le s_{ih},\forall h\in {\mathrm{V}}_i,\forall j\in {\mathrm{P}}_{ih}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \sum _{j\in {\mathrm{V}}_i}{r}_{ijlt}^L\mathbb{I}(s_{ij}<t\le s_{ij}+d_{ij})\le R_{ilt}^L,\forall l\in {\mathrm{L}}_i,\forall t=1,\dots ,T\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \sum _{j\in {\mathrm{V}}_i}{r}_{ijgt}^G\mathbb{I}(s_{ij}<t\le s_{ij}+d_{ij})\le {\overline{R}}_{gt}^G,\forall g\in \mathrm{G},\forall t=1,\dots ,T\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & s_{ij}\in \{A{T}_i,\dots ,T-d_{ij}\},f_{ij}=s_{ij}+d_{ij},\forall j\in {\mathrm{V}}_i.\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & s_{i0}=f_{i0}=A{T}_i.\hfill \end{array}$$

(6)

The activity start times $s_{ij}$ are the scheduling decision variables and determine the resource-use indicators in Constraints (3) and (4). The interval $(s_{ij},f_{ij}]$ follows the numbered-work-period convention used in the implementation and is equivalent to the common $[s_{ij},f_{ij})$ convention under a one-period index shift [42,43].

Let $D_i=A{T}_i+D{D}_i$ denote the absolute due date. For each project $i\in \mathrm{MP}$, the PA adopts one of the following cost-minimization formulations as $OB{J}_i$:

1.

Minimization of the completion time:$minF{T}_i=min{f}_{i,J_i+1}$.

2.

Minimization of the project duration: $minP{D}_i=min\left(F{T}_i-A{T}_i\right)$.

3.

Minimization of the completion time deviation: $minPT{D}_i=min\left|F{T}_i-D_i\right|$.

4.

Minimization of the delay time (Tardiness): $minD{T}_i=minmax\left\{0,F{T}_i-D_i\right\}$.

5.

Minimization of the delay cost: $minD{C}_i=min\left(C_i^{\mathrm{Delay}}D{T}_i\right)$.

Completion time and project duration represent two distinct coordination criteria. Completion time locates project delivery on the common planning timeline, whereas project duration measures the elapsed time from project arrival to completion. Accordingly, they are associated with different project-priority rules and different multi-project aggregation functions.

Upon completion of the initial scheduling, each PA submits its daily global resource demands and schedule metrics (e.g., progress and cost) to the CA. This initiates the global scheduling stage, which allocates global resources.

3.3. Global Scheduling Model

Global scheduling process based on multi-criteria decision-making.

The CA aggregates global resource demands from all projects to identify periods of supply shortage. A final multi-project schedule is globally feasible only if

$$\sum _{i\in \mathrm{MP}}\sum _{j\in {\mathrm{V}}_i}{r}_{ijgt}^G\mathbb{I}(s_{ij}<t\le s_{ij}+d_{ij})\le R_{gt}^G,\forall g\in \mathrm{G},t=1,\dots ,T.$$

(7)

Upon detecting conflicts, the CA initiates resource coordination and triggers rescheduling. Drawing on traditional heuristics that sequence activities by priority, this study proposes a procedure for constructing multi-project schedules using project-level priority rules. Each priority rule yields a multi-project schedule alternative, representing a unique coordination strategy. Subsequently, the CA employs an MCDM method to select the best-ranked schedule from these alternatives based on their attribute values. The procedure for generating a schedule alternative based on a specific priority rule is described below:

1.

Selection of the priority project. Based on the current priority rule, the project with the highest priority is selected. The project schedule remains fixed.

2.

Update of global resource availability. The CA updates the daily availability of global resources by deducting the consumption of the selected project. This updated availability information is then transmitted to the PAs of the remaining projects.

3.

Rescheduling of remaining projects. The PAs of the remaining projects perform local rescheduling based on the updated resource constraints. They then report their revised global resource demands and individual project schedules to the CA.

4.

Aggregation of the multi-project schedule. The CA integrates all individual project schedules to form a complete multi-project schedule alternative and calculates the attribute values at the multi-project level.

Inequity Aversion Theory suggests that stakeholders consider the distribution of individual outcomes when evaluating a collective decision [7]. In operations research, inequity-averse models consider individual outcomes alongside aggregate performance [3]. Based on Inequity Aversion Theory, the proposed mechanism uses Equation (8) to measure the relative loss in each project’s objective value caused by global-resource coordination.

The CA compares each candidate with the initial independent schedule of every project. Let $OB{J}_i\left(S_m\right)$ denote the objective value of project i under schedule $S_m$. $D{C}_i^0$ denotes the delay cost of project i in its initial independent schedule. With ${\left[x\right]}^{+}=max\{0,x\}$, the relative objective deterioration is defined as

$${\delta }_{im}=\left\{\begin{array}{cc}{\displaystyle \frac{{[OB{J}_i\left(S_m\right)-OB{J}_i^0]}^{+}}{P{D}_i^0}},\hfill & OB{J}_i\in \{F{T}_i,P{D}_i,PT{D}_i,D{T}_i\},\hfill \\ {\displaystyle \frac{{[D{C}_i\left(S_m\right)-D{C}_i^0]}^{+}}{D{C}_i^{\mathrm{ref}}}},\hfill & OB{J}_i=D{C}_i.\hfill \end{array}\right.$$

(8)

For the time-based objectives, the numerator and denominator in Equation (8) are both measured in time. For the delay-cost objective, ${[D{C}_i\left(S_m\right)-D{C}_i^0]}^{+}$ is measured in cost. The denominator $D{C}_i^{\mathrm{ref}}$ is also measured in cost and provides a positive project-specific reference total delay cost. In the reported MPSPLIB instances, each project has a constant unit delay cost, giving $D{C}_i\left(S\right)=C_i^{\mathrm{Delay}}D{T}_i\left(S\right)$ and $D{C}_i^{\mathrm{ref}}=C_i^{\mathrm{Delay}}P{D}_i^0$. The delay-cost deterioration is therefore algebraically equivalent to the delay-time deterioration under this setting. When the marginal unit delay cost varies over the delay period, the more general expressions $D{C}_i\left(S\right)={\sum }_{\tau =1}^{D{T}_i\left(S\right)}{c}_{i\tau }^{\mathrm{Delay}}$ and $D{C}_i^{\mathrm{ref}}={\sum }_{\tau =1}^{P{D}_i^0}{c}_{i\tau }^{\mathrm{Delay}}$ provide a cost-based deterioration measure. The resulting ${\delta }_{im}$ is therefore dimensionless for every objective.

Each PA’s initial independent schedule provides the project-specific reference used to calculate ${\delta }_{im}$. The tolerance ${\rho }^{max}$ retains alternatives with acceptable relative loss for every project, and MCDM ranks these resource-feasible alternatives based on their coordinated project outcomes. This sequence translates inequity aversion into project-level loss protection and fairness-aware schedule selection.

Table 3. Project Priority Rules.

Priority Rule	Direction	Priority Index
Earliest Project Completion Time	min	$F{T}_i$
Shortest Project Duration	min	${PD}_i$
Maximum Deviation	max	${PTD}_i$
Maximum Delay Time	max	${DT}_i$
Maximum Delay Cost	max	${DC}_i$

lists the priority rules employed in this study. The five rules correspond to the five project objectives and generate interpretable schedule alternatives.

Multi-Criteria Decision-Making Method. The MCDM method is employed to aggregate the performance of alternatives across multiple criteria, aiming to determine the best compromise solution. The priority rules generate a finite set of resource-feasible candidates, and MCDM compares the candidates that satisfy the relative-deterioration limit. The global scheduling model based on MCDM is formulated as follows:

$$\begin{array}{cc}\hfill \underset{m\in \mathrm{PR}}{max}& \mathrm{EV}\left(S_m\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& {\delta }_{im}\le {\rho }^{max},\forall i\in \mathrm{MP}.\hfill \end{array}$$

(9)

where m denotes the index of a priority rule, and $\mathrm{PR}$ represents the set of candidate priority rules. $S_m$ denotes the multi-project schedule generated by the m-th priority rule. The function $\mathrm{EV}(·)$ maps the multi-dimensional attributes of a schedule to a single comprehensive score. For TOPSIS, $\mathrm{EV}\left(S_m\right)=C_m$, the relative-closeness coefficient calculated in Section 4.3.1. For CoCoSo, $\mathrm{EV}\left(S_m\right)={\mathrm{EV}}_m^{\mathrm{CoCoSo}}$, the comprehensive evaluation value calculated in Section 4.3.2.

Calculation of Multi-Project Schedule Attributes. The CA derives the attributes of the multi-project schedule by aggregating individual project metrics to support multi-criteria decision-making. These attributes, along with their formulations, are detailed in

Table 4. Definition of multi-project schedule attributes.

Attribute	Notation	Formulation	Description
Multi-project makespan	MFT	$\underset{i\in \mathrm{MP}}{max}\left\{F{T}_i\right\}$	Maximum finish time among all projects
Total project duration	MPD	${\displaystyle \sum _{i\in \mathrm{MP}}}P{D}_i$	Sum of individual project durations
Total due-date deviation	MPTD	${\displaystyle \sum _{i\in \mathrm{MP}}}PT{D}_i$	Cumulative deviation from due dates
Total delay time	MDT	${\displaystyle \sum _{i\in \mathrm{MP}}}D{T}_i$	Total accumulated delay time
Total delay cost	MDC	${\displaystyle \sum _{i\in \mathrm{MP}}}D{C}_i$	Total accumulated delay cost

.

4. The Two-Stage Algorithm with MCDM Methods

4.1. Framework of the Two-Stage Scheduling Algorithm

This study proposes an MCDM-based algorithm to solve the two-stage model. The separation preserves each PA’s local activity information and objective while allowing the CA to coordinate only shared-resource demand and schedule-level attributes. It also supports the repeated local rescheduling required after the CA fixes a project in each coordination round. This section introduces the overall framework, followed by a detailed description of the procedures for the local and global scheduling stages.

The pseudocode is presented in Algorithm 1.

Algorithm 1: Two-Stage Multi-Project Scheduling Algorithm

1.

Step 1: Initial Scheduling (Local Scheduling). Each Project Agent (PA) formulates an initial schedule based on its local resource and activity information and the complete global-resource availability. The resulting schedules are individually feasible but may conflict when combined. Subsequently, the PA reports the project’s global resource demands and attributes values to the Coordination Agent (CA).

2.

Step 2: Multi-Alternative Generation (Global Scheduling). The CA aggregates and inspects the total usage of global resources to identify all dates where total demand exceeds supply. These dates are referred to as resource-conflict days. The CA then generates multiple alternatives for global resource scheduling. For each alternative, the CA calculates the remaining available global resources and transmits this information to the respective PAs. It is important to note that within any given alternative, the schedule of the highest-priority project remains unchanged.

3.

Step 3: Rescheduling (Local Scheduling). Upon receiving the global resource availability information for each alternative, each PA other than the highest-priority PA participating in the current negotiation round reformulates its schedule. This process integrates the local resource constraints with the global limits associated with the alternative. The PAs then report their updated global resource demands for each alternative to the CA.

4.

Step 4: Multi-Criteria Decision-Making (Global Scheduling). The CA calculates ${\delta }_{im}$ for each alternative, retains the alternatives satisfying ${\delta }_{im}\le {\rho }^{max}$ for all projects, and applies an MCDM method to select the best-ranked schedule from the retained alternatives. Based on the global resource consumption of the highest-priority project in the chosen alternative, the global resource pool is updated. The PA associated with this project then withdraws from subsequent negotiation rounds. Next, the CA determines whether global resource conflicts persist in the new schedule. If conflicts exist, the daily availability of the remaining global resources serves as the constraint for the next round of global scheduling, and the process returns to Step 2 for the remaining PAs to reschedule. If no resource conflicts remain, the schedule derived from the selected alternative is adopted as the final multi-project schedule.

The framework of the two-stage scheduling algorithm is illustrated in Figure 2. In this algorithm, Reschedule($P{A}_i,{\mathcal{G}}_m$) is the algorithm for initial scheduling and rescheduling in the local scheduling stage. In contrast, MCDM($A_1^m,\dots ,A_k^m$) is the MCDM algorithm for the global scheduling stage. The two-stage approach using TOPSIS for global coordination is denoted TA-TOPSIS, whereas its CoCoSo-based counterpart is denoted TA-CoCoSo. The global benchmark is the distributed multi-agent scheduling/auction-based negotiation mechanism (DMAS/ABN), while the local CGA is compared with a standard genetic algorithm (GA) and particle swarm optimization (PSO).

The residual-capacity update provides a direct link between the local feasibility constraints and the complete global-resource capacity constraint. Provided that each local rescheduling problem is feasible and the retained candidate set is nonempty, the procedure preserves feasibility across coordination rounds and returns a multi-project schedule satisfying Constraints (2)–(6) and the global-resource capacity constraint (7). Appendix A provides the formal derivation.

4.2. Local Scheduling Algorithm

The local scheduling stage addresses the single-project scheduling problem, which is known to be NP-hard [44]. To address large-scale instances effectively, meta-heuristic algorithms are widely adopted. Given that the two-stage scheduling approach frequently requires generating single-project schedules, this study proposes a Chaotic Genetic Algorithm (CGA) integrated with a Serial Schedule Generation Scheme (SSGS) to enhance search efficiency. The method supports repeated local schedule generation and is evaluated against GA and PSO.

For repeated local schedule generation, the standard elitist GA is enhanced with an adaptive permutation-mutation mechanism. The mutation probability is jointly adjusted according to evolutionary progress and population fitness variance, while a Logistic-map value selects among swap, insertion, and inversion operators. This mutation mechanism mitigates premature convergence by maintaining variation in activity-priority permutations under elitist selection.

The model represents schedules through the activity start times $s_{ij}$. A CGA chromosome specifies the priority ranks of real activities, and SSGS decodes these ranks into activity start and finish times. During decoding, SSGS checks period-specific resource feasibility over each activity’s execution interval $s_{ij}<t\le s_{ij}+d_{ij}$.

The Schedule Generation Scheme (SGS) is a procedure for incrementally constructing a complete schedule while ensuring feasibility at every iteration. It classifies project activities into three sets: the scheduled set, the eligible set, and the ineligible set. Activities enter the eligible set after all predecessors have been scheduled. During the scheduling process, an activity is selected from the eligible set based on its priority, assigned the earliest resource-feasible start time, and moved to the scheduled set. SGS methods are categorized into Serial SGS (SSGS) and Parallel SGS (PSGS). Although PSGS searches a smaller solution space, it may exclude optimal solutions [45]. Consequently, this study employs the SSGS.

The encoding details of the CGA based on SSGS are as follows. Each gene corresponds to the priority of an activity, and the sequence of these priorities forms a chromosome. Specifically, the real activities receive a permutation of the integer ranks $1,\dots ,J_i$, where $J_i$ denotes the number of non-dummy activities in project i. A smaller value indicates a higher priority. The priorities of the dummy start and end activities are fixed at 0 and $J_i+1$, respectively.

The workflow of the algorithm is presented in Algorithm 2 and summarized as follows:

1.

Initialization: A population of chromosomes is randomly generated.

2.

Decoding: The SSGS transforms each chromosome into a feasible scheduling scheme.

3.

Evaluation: The objective function value is adopted as the fitness value.

4.

Evolutionary Operations:

• Selection:Tournament selection is applied. Individuals compete in pairs, and the higher-fitness individuals are retained.
• Crossover: Partially matched crossover (PMX) is performed with crossover probability $p_c$. Two crossover positions delimit the exchanged segments, and the resulting element mappings are used to remove duplicate activity ranks outside the exchanged segments. The offspring therefore remain valid activity-priority permutations.
• Mutation: Adaptive chaotic mutation is applied with a generation-specific probability $p_g$. A Logistic-map state selects swap, insertion, or inversion after mutation is triggered.

5.

Population Update: An elitist strategy is used to merge parent and offspring populations, preserving the top-ranking individuals for the next generation.

6.

Termination: Steps 2 to 5 are repeated until either the maximum number of iterations or the maximum stagnation generation count is reached.

Algorithm 2: Chaotic Genetic Algorithm for Local Scheduling

The adaptive mutation probability is determined from the evolutionary progress and the dispersion of population fitness. Let ${\varphi }_{ig}$ denote the scaled fitness of individual i in generation g, and let $v_g$ denote the corresponding population fitness variance.

$$v_g=Var\left({\varphi }_{1g},\dots ,{\varphi }_{Ng}\right).$$

The bounded diversity factor $d_g$ is defined as

$$d_g=min\left\{d_{max},max\left\{d_{min},v_g\right\}\right\}.$$

The mutation probability in generation g is then calculated as

$$p_g=min\left\{p_{max},max\left\{p_{min},p_m^0\left(1-\alpha {\displaystyle \frac{g}{G_{max}}}\right)d_g\right\}\right\}.$$

Here, $p_m^0$ is the base mutation probability, $\alpha$ is the decay coefficient associated with evolutionary progress, $d_{min}$ and $d_{max}$ are the bounds on the diversity factor, and $p_{min}$ and $p_{max}$ are the bounds on the mutation probability. These parameters satisfy $0\le \alpha \le 1$, $0<d_{min}\le d_{max}$, and $0<p_{min}\le p_{max}<1$.

Let h denote the current evolutionary iteration, where $h=1,\dots ,G_{max}$. A Logistic state $x_{i,0}\in (0,1)$ is initialized for each population position i before the evolutionary iterations. At iteration h, the retained state is updated according to

$$x_{i,h}=\mu x_{i,h-1}\left(1-x_{i,h-1}\right),$$

Here, $\mu$ is the Logistic-map parameter. The updated value $x_{i,h}$ is retained for the next iteration and is used to select the mutation operator in the current iteration. The index h corresponds to the generation counter g in Algorithm 2. Let ${\tau }_1$ and ${\tau }_2$ be operator-selection thresholds satisfying $0<{\tau }_1<{\tau }_2<1$. After mutation is triggered with probability $p_g$, swap is selected when $x_{i,h}<{\tau }_1$, insertion is selected when ${\tau }_1\le x_{i,h}<{\tau }_2$, and inversion is selected when $x_{i,h}\ge {\tau }_2$. Swap exchanges two distinct positions, insertion moves the gene at one selected position to another selected position, and inversion reverses the genes within a selected interval. Each operation preserves the permutation encoding.

4.3. Global Scheduling Algorithm

This section uses TOPSIS and CoCoSo to rank feasible coordination alternatives generated in the global stage. Both methods use the same cost-oriented attributes, with CoCoSo providing a comparative MCDM ranking.

Table 5. Decision characteristics of TOPSIS and CoCoSo.

Method	TOPSIS	CoCoSo
Ranking Basis	Determines ranking based on the Euclidean distance of each alternative from the positive ideal and negative ideal solutions.	Integrates the Weighted Sum and Weighted Product models, employing a compromise coefficient to formulate a unified solution.
Compensation	Weak compensation: Offsetting extreme disadvantages solely through Euclidean distance measures is challenging.	Structured compromise: Balances complete and weak compensation mechanisms to provide a middle-ground solution.
Compromise emphasis	Adheres to the weakest link principle, where the performance of an alternative is constrained by its poorest criterion.	Balances the maximization of utility with the avoidance of weakest links.
Tendency	Prefers alternatives with no significant weaknesses.	Prefers robust solutions, thereby mitigating the impact of extreme values.

illustrates the differences between these two methods.

MCDM methods typically require relative attribute weights. Multiple-attribute group decision making aligns member aggregation and criterion weighting with the decision setting and the information provided by participants [46]. Equal participant weights provide a conventional aggregation specification when participants have equal coordination rights [47]. In the present model, each PA adopts one single-project objective and contributes one equal representation unit. Attribute n therefore has $f_n$ such units, while the portfolio contains ${\sum }_{l\in ATR}{f}_l$ units across all attributes. Equation (10) assigns attribute n its share of equal PA representation in the project portfolio. Frequency-normalized criterion weights have also been used in crowd decision making, where criterion weights reflect their relative evaluation frequencies [48]. The weight ${\omega }_n$ for attribute n is calculated as presented in Equation (10).

$$\omega =\left[{\omega }_1,{\omega }_2,\dots ,{\omega }_N\right]=\left[{\displaystyle \frac{f_1}{{\sum }_{n\in ATR}{f}_n}},{\displaystyle \frac{f_2}{{\sum }_{n\in ATR}{f}_n}},\dots ,{\displaystyle \frac{f_N}{{\sum }_{n\in ATR}{f}_n}}\right].$$

(10)

To support rapid weight acquisition and automatic calculation during repeated coordination, the proposed algorithm derives the weight vector from the frequency with which each objective is adopted. For a general coordination instance with $f_n=0$, Equation (10) gives ${\omega }_n=0$. In TOPSIS, the normalization formula remains unchanged. The zero weight gives $z_{mn}=0$ for every alternative, so the attribute contributes zero to both ideal-solution distances. If an unused attribute is present in a CoCoSo input, it is removed before calculation to avoid a zero-exponent term in the power-weighted component.

After the project-level screen, each feasible alternative is represented by the schedule-level attributes MFT, MPD, MPTD, MDT, and MDC. Let $b_{mn}$ denote attribute n in alternative m, and let $r_{mn}$ denote its normalized value. Each attribute column is transformed into dimensionless values before evaluation. TOPSIS uses vector normalization. CoCoSo uses minimum and maximum normalization because lower values are preferred for all attributes. The normalized values are aggregated into the method-specific $\mathrm{EV}\left(S_m\right)$, which determines the ranking of the feasible alternatives.

4.3.1. TOPSIS Method-Based Global Scheduling Algorithm

The Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) ranks multi-project scheduling alternatives based on their relative proximity to an ideal solution. In this problem, the positive ideal combines the lowest observed value of every scheduling cost attribute, whereas the negative ideal combines the highest observed values. The distances therefore show how close each complete coordination alternative is to the best attainable attribute profile within the current candidate set. Widely recognized in engineering domains for its intuitive decision-making logic [49], the TOPSIS-based global scheduling procedure is summarized in Algorithm 3 and detailed below.

1.

Normalization: The decision matrix is normalized using Equation (11).

$$r_{mn}={\displaystyle \frac{{\overline{b}}_{mn}}{\sqrt{{\displaystyle \sum _{m\in \mathrm{PR}}}{\left({\overline{b}}_{mn}\right)}^2}}}$$

(11)

2.

Weighted Matrix Construction: The weighted decision matrix $Z={\left[z_{mn}\right]}_{M\times N}$ is computed, where $z_{mn}={\omega }_n·r_{mn}$.

3.

Ideal Solution Determination: The positive ideal solution ($Z^{+}$) and the negative ideal solution ($Z^{-}$) are identified. $Z^{+}$ comprises the lowest cost value for each attribute across all alternatives, while $Z^{-}$ consists of the highest cost values. All attributes are cost criteria, so these are defined in Equation (12) and Equation (13), respectively.

$$\begin{array}{cc}\hfill Z^{+}& ={\left[z_n^{+}\right]}_{1\times N}=\left[\underset{m\in \mathrm{PR}}{min}{z}_{m1},\underset{m\in \mathrm{PR}}{min}{z}_{m2},\dots ,\underset{m\in \mathrm{PR}}{min}{z}_{mN}\right]\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill Z^{-}& ={\left[z_n^{-}\right]}_{1\times N}=\left[\underset{m\in \mathrm{PR}}{max}{z}_{m1},\underset{m\in \mathrm{PR}}{max}{z}_{m2},\dots ,\underset{m\in \mathrm{PR}}{max}{z}_{mN}\right]\hfill \end{array}$$

(13)

4.

Distance Calculation: The separation of alternative m from the positive and negative ideal solutions is measured using Euclidean distance, as shown in Equations (14) and (15).

$$\begin{array}{c}\hfill D_m^{+}={\parallel Z_m-Z^{+}\parallel }_2=\sqrt{\sum _{n=1}^N{\left(z_{mn}-z_n^{+}\right)}^2}\end{array}$$

(14)

$$\begin{array}{c}\hfill D_m^{-}={\parallel Z_m-Z^{-}\parallel }_2=\sqrt{\sum _{n=1}^N{\left(z_{mn}-z_n^{-}\right)}^2}\end{array}$$

(15)

5.

Relative Closeness Computation: The relative closeness coefficient is calculated via Equation (16). This metric synthesizes the proximity of an alternative to the positive ideal and its remoteness from the negative ideal. Consequently, multi-project alternatives are ranked in descending order of this value, with the maximum value identifying the best-ranked alternative.

$$C_m={\displaystyle \frac{D_m^{-}}{D_m^{+}+D_m^{-}}}$$

(16)

Algorithm 3: TOPSIS-Based Multi-Project Decision

4.3.2. CoCoSo Method-Based Global Scheduling Algorithm

Characterized by diverse compromise mechanisms, the CoCoSo method serves as a robust decision-making tool. It integrates the linear Weighted Sum Method (WSM) and the non-linear power-weighted component customarily denoted WPM to establish a balanced compromise. For the scheduling alternatives, $K^{\left(1\right)}$ reflects the candidate’s share of the combined score, $K^{\left(2\right)}$ compares it with the weakest baseline scores, and $K^{\left(3\right)}$ balances the two components through ${\lambda }^{co}$. Their final aggregation provides a second ranking of the same finite candidate set. This mechanism renders the final rankings insensitive to minor data fluctuations, thus enhancing robustness [50]. Given the complexity of multi-agent decision-making in project scheduling, a method ensuring result stability is essential. Consequently, CoCoSo is well-suited to the DRCMPSP. Algorithm 4 details the global scheduling steps based on CoCoSo.

Algorithm 4: CoCoSo-Based Multi-Project Decision

5. Case Study: Multi-Project Scheduling for UAV Series R&D

This section presents a multi-project scheduling case study that motivates the research and provides a practical context for evaluating the proposed algorithm. A commercial company is developing four unmanned aerial vehicle (UAV) models in parallel for different application scenarios. All four projects follow the same R&D stages: requirements analysis, system design, prototype manufacturing, and integration testing. However, their performance requirements are different. Four separate teams carry out the projects. Each project relies on local resources, such as flight-control and structural design engineers. These resources are scheduled within each project rather than across projects. For brevity, we report only the information relevant to project scheduling. Brief descriptions of the four projects are given below:

• Project 1: Surveying and mapping model. This project demands high flight control stability and data processing capabilities. The client prioritizes the earliest possible completion. The critical path duration is 91 days, the earliest start time is day 0, and the tardiness cost is 42.
• Project 2: Logistics and transport model. The R&D focus lies on the airframe structure and the battery management system. The project manager aims to minimize the project duration. The critical path duration is 121 days, the earliest start time is day 7, and the tardiness cost is 38.
• Project 3: Industrial inspection model. This project emphasizes long endurance and wind resistance performance. The project manager seeks to minimize tardiness. The critical path duration is 110 days, the earliest start time is day 7, and the tardiness cost is 38.
• Project 4: Agricultural plant protection model. This requires specialized software for flight path planning. The investors and managers aim to minimize delay costs. The critical path duration is 97 days, the earliest start time is day 14, and the tardiness cost is 33.

However, the execution of all four projects depends on a shared global resource: an external public testing and experimental platform for environment and reliability verification. All prototypes, from components to complete assemblies, must reserve testing services on this platform for assessments of environmental adaptability, structural integrity, and interference. The platform’s certified engineers, precision equipment, and available testing slots form a fixed and scarce service capacity. Consequently, the allocation of this global resource constitutes the primary bottleneck in global scheduling. Based on platform availability and engineer-equipment shifts, the platform provides a daily capacity of 28 units, measured in engineer-equipment hours.

Figure 3a shows the initial global resource demand. Each block in the figure is labeled by the project index followed by the activity index. These initial schedules were generated via local scheduling using heuristic algorithms, without accounting for global resource-sharing constraints. As shown in the figure, the cumulative resource demand from Projects 1 to 3 exceeds the available global resource capacity beginning at $t=7$. This resource conflict necessitates initiating the global scheduling stage. The procedure for the first global scheduling iteration is described as follows.

The individual projects aim to minimize completion time, duration, delay time, and delay cost. Accordingly, the coordination agent generates four alternative schemes using priority rules, adopting multi-project objectives corresponding to these individual goals as attribute values. In each scheme, the schedule of the highest-priority project is fixed, whereas the schedules of the other projects are rescheduled based on residual resources. In this illustration, the relative-deterioration limit is set to ${\rho }^{max}=1.0$ to retain a nonempty candidate set for the case data. The value serves as a feasibility setting for this illustrative coordination process. The threshold ${\rho }^{max}$ is a positive ratio with an admissible range of $(0,+\infty )$. A lower value represents stronger inequity aversion through tighter project-level loss protection, while a higher value reflects greater tolerance for objective deterioration. In an applied setting, stakeholders specify the threshold before coordination according to each sub-project’s tolerance for objective loss. The aggregated results form the initial evaluation matrix (

Table 6. Evaluation matrix.

	MFT	MPD	MDT	MDC
Alternative 1	157	492	122	4840
Alternative 2	157	492	122	3471
Alternative 3	146	456	86	3427
Alternative 4	146	455	85	3471

).

The matrix is normalized via TOPSIS to eliminate dimensional differences:

$$\left[\begin{array}{cccc}0.5178& 0.5189& 0.5791& 0.6287\\ 0.5178& 0.5189& 0.5791& 0.4509\\ 0.4815& 0.4809& 0.4082& 0.4452\\ 0.4815& 0.4799& 0.4034& 0.4509\end{array}\right]$$

Assigning equal weights to all attributes, the positive ($Z^{+}$) and negative ($Z^{-}$) ideal solutions are derived as:

$$\begin{array}{cc}\hfill Z^{+}& =[0.1204,0.1200,0.1009,0.1113]\hfill \\ \hfill Z^{-}& =[0.1295,0.1297,0.1448,0.1572]\hfill \end{array}$$

The relative closeness values are calculated as $[0,0.492,0.981,0.978]$, identifying Scheme 3 as the best-ranked alternative. This scheme prioritizes Project 2. Iterative conflict resolution based on Scheme 3 yields the final schedule (Figure 3b). Conversely, a traditional single-criterion approach yields the schedule shown in Figure 3c, resulting in excessive delays for Project 1.

To illustrate the coordination process in a larger project portfolio, Supplementary Material File S2 presents a ten-project example.

6. Numerical Experiment

This section evaluates the performance of the proposed two-stage scheduling algorithm. The algorithms were implemented in Python 3.10.14 and executed on a PC with a 5.2 GHz CPU and 16 GB of RAM. Section 6.1 details the experimental design and instance parameters. Section 6.2 compares the DRCMPSP performance under different decision rules and analyzes the results grouped by problem characteristics. The empirical evaluation covers 105 MPSPLIB instances across nine problem subsets, including three activity scales, three project counts, and both low- and high-resource-contention conditions.

6.1. Experimental Design

The test instances are from the Multi-Project Scheduling Problem Library (MPSPLIB) [51,52], which includes problem instances representing realistic scenarios with varying task sizes. As this study focuses on resource coordination among numerous projects, groups containing only two parallel projects are excluded.

Table 7. Instance descriptions.

Problem Subset	Number of Instances	Number of Projects	Number of Activities	Problem Size	Average OLF
MP30_5	5	5	30	150	0.826
MP30_10	5	10	30	300	2.380
MP30_20	5	20	30	600	3.370
MP90_5	15	5	90	450	3.532
MP90_10	15	10	90	900	2.949
MP90_20	15	20	90	1800	2.009
MP120_5	15	5	120	600	2.974
MP120_10	15	10	120	1200	2.383
MP120_20	15	20	120	2400	2.723

details the instances employed in this research. The MPSPLIB provides the Overload Factor (OLF) for each instance, indicating the degree of resource conflict. An OLF value exceeding 1 signifies intense resource contention. Furthermore, each MPSPLIB instance file specifies global resource indices, daily global resource capacities, project arrival times, and the names of the associated single-project instances. These single-project instances contain information regarding resource requirements, daily local resource capacities, and precedence relations.

The five minimization objectives are assigned to projects in project order. Projects 1 through 5 minimize completion time, delay time, project duration, delay cost, and completion-time deviation, respectively; for instances with more than five projects, the objective sequence is repeated cyclically. Consequently, every attribute is adopted by at least one project in each of the 105 reported instances, so $f_n>0$ throughout the reported experiments. In the reported experiments, cyclic objective assignment gives each objective the same adoption frequency and produces uniform attribute weights.

The deterioration thresholds ${\rho }^{max}$ for MP30_5, MP30_10, and MP30_20 were 1, 3, and 6, respectively. The corresponding values for MP90_5, MP90_10, and MP90_20 were 4, 6, and 9. The values for MP120_5, MP120_10, and MP120_20 were 4, 8, and 9. These values were determined through preliminary runs and fixed for the reported experiments.

6.2. Comparison of Algorithm Performance

Since the two-stage algorithm comprises global and local scheduling stages, we compared the results of different algorithms for each phase separately. Based on Liu et al. [12] and preliminary experiments, the Chaotic Genetic Algorithm (CGA) with elitism was configured with a maximum of 200 generations, a population size of 50, and a crossover probability of $p_c=0.7$. The adaptive-mutation parameters were set to $p_m^0=0.9$, $\alpha =0.7$, $d_{min}=0.5$, $d_{max}=2.0$, $p_{min}=0.01$, and $p_{max}=0.8$. The Logistic-map parameter and operator-selection thresholds were set to $\mu =4$, ${\tau }_1=0.3$, and ${\tau }_2=0.6$, respectively. The algorithm terminates upon reaching the maximum number of iterations or when the objective function shows no improvement for 30 consecutive generations. For global scheduling, the compromise coefficient ${\lambda }^{co}$ in the CoCoSo method was set to 0.5.

6.2.1. Comparison of Global Scheduling Algorithms

We evaluated the performance of the proposed two-stage algorithm with a TOPSIS selection mechanism (TA-TOPSIS) against the DMAS/ABN mechanism [53]. DMAS/ABN is an established benchmark for comparing coordination mechanisms in DRCMPSP. It was originally developed for cash-flow-oriented coordination. This objective difference may affect the interpretation of its performance on the fairness-oriented indicators. The comparison also illustrates the contribution of the proposed method because MCDM supports fairness-aware distributed coordination. Furthermore, to benchmark TOPSIS against other multi-criteria decision methods, we replaced TOPSIS with the CoCoSo method [50], yielding a two-stage algorithm with CoCoSo (TA-CoCoSo).

Game-theoretic coordination in DRCMPSP requires model-specific strategy spaces, payoff functions, information structures, interaction sequences, and stopping conditions [17,37]. The current numerical tests do not include game-theoretic baselines because a controlled comparison requires these elements to be defined consistently under the same project objectives, available information, and coordination outputs. The proposed MCDM mechanism operates at the alternative-selection stage. Once the priority rules have generated a finite set of complete resource-feasible schedules and the criteria and weights have been specified, TOPSIS and CoCoSo determine the schedule ranking through a finite sequence of matrix calculations. For the coordination task studied here, this finite and deterministic procedure constitutes a theoretical advantage over game-theoretic coordination because it determines the schedule ranking without requiring equilibrium existence or the convergence of iterative strategic learning.

Focusing solely on the total delay time is inadequate due to the heterogeneity of project objectives.

Table 8. Performance comparison over different subsets.

	Mean				Std				Max
Subset	TOPSIS	DMAS/ABN	CoCoSo	Gap	TOPSIS	DMAS/ABN	CoCoSo	Gap	TOPSIS	DMAS/ABN	CoCoSo	Gap
MP30_5	57.0	73.5	53.9	5.8%	48.2	78.2	35.0	37.7%	141.5	218.2	111.0	27.5%
MP30_10	180.6	200.9	202.3	0	250.1	289.0	277.4	0	840.8	962.2	895.0	0%
MP30_20	470.4	439.7	447.2	7.0%	774.4	694.1	761.7	11.6%	2993.8	2603.6	3132.4	15.0%
MP90_5	1011.3	1186.7	1202.9	0	1628.3	1966.9	2014.8	0	4244.3	5105.1	5219.7	0%
MP90_10	1013.5	1181.1	1268.3	0	2120.5	2674.2	2843.4	0	7188.2	9073.4	9492.7	0%
MP90_20	564.3	577.0	527.2	6.0%	1143.1	1187.7	1143.2	0	4313.9	4412.5	4683.5	0%
MP120_5	3295.7	3549.8	3753.2	0	6249.1	6772.4	7181.8	0	15,789.4	17,091.1	18,112.4	0%
MP120_10	1138.7	1240.7	1228.9	0	2522.7	2857.0	2701.3	0	8267.5	9526.7	8739.6	0%
MP120_20	1426.8	1519.4	1377.1	3.6%	3656.4	3905.1	3657.6	0	14,939.0	15,829.9	15,432.9	0%

Note: Bold values denote the best results for each group across all considered algorithms.

presents the mean, standard deviation, and maximum values for each problem subset.

Table A1. Dispersion of relative deterioration across problem subsets.

Subset	TA-TOPSIS	DMAS/ABN	TA-CoCoSo
MP30_5	0.1601	0.2175	0.1656
MP30_10	0.9414	0.9043	0.8795
MP30_20	1.5423	1.6056	1.6086
MP90_5	0.6991	0.6341	0.6621
MP90_10	1.0193	1.3204	1.0289
MP90_20	0.7061	0.8517	0.7136
MP120_5	0.6435	0.8621	0.6161
MP120_10	0.8155	0.9454	0.8039
MP120_20	1.0438	1.1775	1.0713

Note: Values are the subset averages of the instance-level population standard deviations calculated across project-level relative deterioration values.

in Appendix B additionally reports the dispersion of dimensionless relative deterioration values for the three coordination methods. Here, Gap denotes the percentage deviation of the TA-TOPSIS solution from the best reported value. Specifically, in the MP30_20 subset, characterized by fewer activities and high project parallelism, the performance gap between TA-TOPSIS and the auction-based method does not exceed 15%.

However, applicability varies across MCDM methods. TA-TOPSIS has lower standard-deviation and maximum values in several subsets, whereas TA-CoCoSo yields lower mean values in some subsets. These descriptive differences are assessed with the Wilcoxon tests reported below.

Figure 4 illustrates the pairwise performance gaps between the algorithms. The x-axis represents the instance subsets, labeled as MP$N_A$_$N_P$, where $N_A$ and $N_P$ denote the numbers of activities and projects, respectively.

Based on the OLF degrees, the instances are categorized into two groups representing high and low global resource contention. Figure 5 illustrates the performance trends relative to instance size. Figure 6 provides the corresponding descriptive high-contention patterns by activity and project count.

Table 9. Wilcoxon signed-rank test results comparing TA-TOPSIS against DMAS/ABN and CoCoSo.

OLF Degree	Metric	No. of Inst.	vs. DMAS/ABN		vs. CoCoSo
			p-Value	Sig.	p-Value	Sig.
ALL	Mean	105	0.0124	*	0.0431	*
	Std	105	0.0217	*	0.0325	*
	Max	105	0.0566	ns	0.0300	*
HIGH	Mean	80	0.0141	*	0.0123	*
	Std	80	0.0073	**	0.0067	**
	Max	80	0.0151	*	0.0088	**
LOW	Mean	25	0.7103	ns	0.0768	ns
	Std	25	0.3458	ns	0.0424	*
	Max	25	0.1454	ns	0.1592	ns

Note: * $p<0.05$; ** $p<0.01$; ns: not significant.

presents the results of the Wilcoxon signed-rank tests for pairwise comparisons of the global scheduling algorithms.

For high contention, the tests support differences in the mean, standard deviation, and maximum values against both comparators. Under low contention, the standard-deviation comparison with TA-CoCoSo yields $p=0.0424$, while the remaining comparisons yield $p>0.05$. TA-TOPSIS is subsequently used as the coordination mechanism in the local-scheduling comparison. The stratified Wilcoxon results show that the standard-deviation difference remains statistically significant across the 80 high-contention instances against both comparators, supporting the stability of the principal outcome-balance finding under pronounced resource contention.

To provide a centralized reference for assessing the efficiency loss associated with distributed coordination, an additional small-scale comparison was conducted using instances in the MP30_5 subset of MPSPLIB. The model specification, efficiency-loss calculation, and complete results are provided in Supplementary Material File S1.

6.2.2. Comparison of Local Scheduling Algorithms

The proposed CGA, which incorporates elitism and adaptive permutation mutation, is compared with a standard Genetic Algorithm (GA) and particle swarm optimization (PSO). The Wilcoxon test results indicate that the CGA possesses a significant computation-time advantage in some high-contention instance groups. Figure 7 displays the boxplots of runtime differences for instance subsets grouped by problem size. The vertical axis reports the runtime difference $A-B$, where A and B denote the CGA and the standard GA, respectively. The figure reveals that the average runtime differences consistently fall below the zero line, particularly for problem sizes greater than or equal to 600. This indicates that the CGA is well-suited to large-scale scheduling problems.

The instances were further categorized by OLF degree and number of activities.

Table 10. Runtime of local scheduling and Wilcoxon signed-rank test results.

OLF	Number of Activities	Run Time of GA	Run Time of PSO	Run Time of CGA	Gap	Significance
HIGH	30	575.6	602.3	581.1	0.95%	ns
HIGH	90	1944.6	1937.4	1893.3	0	**
HIGH	120	3856.0	4232.6	3740.7	0	**
LOW	30	96.9	101.2	97.7	0.8%	ns
LOW	90	1092.6	1165.2	1091.9	0	ns
LOW	120	5556.2	5442.9	5387.7	0	ns

Note: Bold values denote the best results for each group across all considered algorithms. ** $p<0.01$; ns: not significant.

reports the mean runtime and the results of the Wilcoxon signed-rank test for each subset. The findings show that the proposed CGA achieves a significant runtime advantage when global resource contention is high and the number of activities is at least 90. In all other groups, the algorithm demonstrates either a marginal advantage or a performance difference of less than 1%.

7. Conclusions

This study addresses the Distributed Resource-Constrained Multi-Project Scheduling Problem (DRCMPSP), where autonomous projects with heterogeneous objectives compete for shared resources in distributed settings. To mitigate the inefficiency and perceived unfairness caused by resource contention, we propose the two-stage algorithm with a TOPSIS selection mechanism (TA-TOPSIS). Specifically, a coordination mechanism based on Multi-Criteria Decision-Making (MCDM) is designed, employing the TOPSIS method to evaluate competing resource requests objectively. Furthermore, the local CGA enhances elitist genetic search with adaptive permutation mutation to improve the computational efficiency of local schedule generation. Computational experiments using MPSPLIB instances and a UAV R&D case study validate the effectiveness of the proposed approach.

Under high resource contention, TA-TOPSIS produces significantly more balanced project-level outcomes than both DMAS/ABN and TA-CoCoSo. Its lower standard deviation reflects a more even distribution of project-level outcomes in the evaluated high-contention instances. The comparison with TA-CoCoSo also reveals a complementary trade-off. TA-TOPSIS places greater emphasis on outcome balance, whereas TA-CoCoSo can attain lower mean objective values in some problem settings. Accordingly, TA-TOPSIS is better suited to coordination settings in which outcome balance is a primary concern, while TA-CoCoSo remains useful when managers place greater emphasis on the overall objective level. The stronger balance advantage under high contention further indicates that explicit multi-criteria coordination becomes more valuable as resource competition intensifies and priority decisions have greater consequences for individual projects. At the local stage, the runtime results show that the enhanced CGA can support the repeated rescheduling required by the two-stage coordination process.

From a managerial perspective, this study shows that fairness considerations can be incorporated into distributed multi-project coordination by explicitly comparing heterogeneous project outcomes. Unlike centralized approaches that enforce a single global objective, the proposed MCDM-based mechanism provides a transparent and justifiable basis for resource allocation that respects the autonomous yet competing nature of projects. This approach makes the balance of project-level outcomes explicit while maintaining the autonomy of individual projects, thereby supporting coordinated management of the project portfolio.

Several limitations of the current study point to directions for future research. The proposed method currently relies on deterministic parameters and static decision weights. Future work will extend the model to stochastic environments to address uncertainties in activity durations and resource availability. Additionally, exploring adaptive weight adjustment mechanisms and parallel computing architectures offers avenues to further enhance both decision robustness and computational efficiency for large-scale DRCMPSP instances. A common payoff specification and interaction protocol can also support direct comparison with game-theoretic coordination mechanisms.