Efficiency and Fairness in Physical Internet Logistics Coordination Under Shared Capacity Constraints

Abstract

Background: The Physical Internet (PI) promotes resource sharing among independent firms. This can improve logistics efficiency, but shared route capacity and limited compensation may also create unequal outcomes among firms. Methods: This study develops a framework for coordinated logistics planning under shared route capacity constraints. The framework includes two coordination rules. Model 3.3 is an efficiency-oriented participation-guaranteeing rule with individual rationality constraints. Model 3.4 is a fairness-oriented rule that minimizes the maximum firm-level disadvantage under a limited compensation budget. Numerical experiments are conducted using a stylized Japanese domestic consumer goods distribution network. Results: Coordinated planning reduces total logistics cost compared with decentralized sequential allocation. Model 3.3 achieves the lowest system cost but gives benefits unevenly. Model 3.4 gives more balanced firm-level outcomes and improves the worst-off firm in the tested scenarios. The results also show that substantial fairness improvements can be obtained through small route-allocation changes. Conclusions: The study shows how two coordination rules can be used in PI-oriented logistics coordination. Model 3.3 is useful when firms need a no-loss guarantee, especially in an early stage. Model 3.4 is useful in a mature or repeated coordination stage, where the platform needs to avoid excessive disadvantage.

1. Introduction

In recent years, the concept of the Physical Internet (PI) has attracted increasing attention as a new paradigm for designing open and interconnected logistics networks. The PI aims to improve the efficiency and sustainability of freight transportation by enabling the sharing of logistics resources among multiple independent companies. Through standardized containers, shared transportation infrastructure, and digital coordination platforms, goods can move through logistics networks in a manner similar to data transmission on the Internet.

However, coordination in open logistics networks faces significant challenges. When multiple firms share transportation resources such as routes, vehicles, or terminals, capacity limitations may lead to competition among participants. If each firm optimizes its own transportation decisions independently, the resulting allocation of resources may be inefficient at the system level. Coordinated planning can reduce total logistics costs and improve resource utilization, but it may also create unequal outcomes among firms.

To encourage participation in coordinated logistics systems, compensation mechanisms are often introduced. Firms that experience loss under coordinated plans may receive financial compensation so that no participant becomes worse off than in its standalone operation. Such mechanisms are commonly used in collaborative logistics and shared transportation systems to ensure voluntary participation and maintain system stability.

Nevertheless, compensation mechanisms may not always be sufficient. In practice, compensation resources may be limited, especially in early-stage collaborative logistics platforms or pilot coordination initiatives. In shared logistics systems, uneven resource allocation often leads to unfair benefit distribution among firms. Therefore, it is important to explore how fairness can be improved under a limited compensation budget.

Accordingly, it is necessary to consider not only system efficiency but also fairness among participants. While efficiency-oriented coordination focuses on minimizing the total logistics cost of the system, fairness-oriented coordination aims to reduce the inequality of cost changes among participating firms. In particular, minimizing the maximum disadvantage among firms can provide a balanced solution.

Motivated by these challenges, this study develops an optimization framework for coordinated logistics planning in open logistics networks. The framework considers multiple firms sharing limited transportation resources and incorporates compensation mechanisms. In addition to an efficiency-oriented coordination model that minimizes the total system cost, a fairness-oriented min–max model is introduced that minimizes the maximum disadvantage experienced by any participating firm.

The contribution of this study does not lie in proposing a fundamentally new fairness algorithm itself. Instead, this study proposes a PI-oriented coordination framework with two complementary rules under shared transportation capacity and limited compensation resources. Model 3.3 is an efficiency-oriented participation-guaranteeing rule. Model 3.4 is a fairness-oriented rule that balances the worst firm-level disadvantage. This is different from studies that discuss fairness only after costs or profits have already been allocated. In this study, fairness is linked directly to route-allocation decisions. The two rules can also be used at different stages of platform development. Model 3.3 is useful when firms first join the platform and need a no-loss guarantee. Model 3.4 is useful when coordination becomes more stable, and the platform wants to prevent excessive or repeated disadvantage.

The main contributions of this study are summarized as follows:

• A coordinated logistics planning framework is developed for multi-player logistics networks under shared transportation capacity constraints.
• The role of compensation mechanisms in ensuring voluntary participation in coordinated logistics systems is analyzed.
• A two-rule coordination framework is introduced. Model 3.3 guarantees that no firm becomes worse off after compensation. Model 3.4 reduces the worst firm-level disadvantage when this strict guarantee is difficult to provide in every period because compensation resources are limited.

The remainder of this paper is organized as follows: Section 2 reviews related studies on logistics coordination and fairness-oriented optimization. Section 3 presents the coordinated logistics planning model and introduces the fairness-oriented min–max formulation. Section 4 provides numerical experiments based on a representative Japanese domestic consumer goods distribution network. Section 5 discusses the theoretical, practical, and policy implications of the study, as well as its limitations and future research directions. Section 6 concludes the paper.

2. Literature Review

2.1. Coordination and Cooperation in Logistics Systems

Logistics coordination has long been recognized as an important approach for improving supply chain performance. In traditional supply chains, coordination mechanisms are often implemented through contractual agreements among partners, such as revenue-sharing or cost-sharing contracts. These mechanisms aim to align incentives among supply chain actors and improve overall system efficiency. For example, revenue-sharing contracts have been widely studied as an effective coordination mechanism that can reduce double marginalization and improve supply chain performance (Cachon and Lariviere, 2005 [1]; Bart et al., 2021 [2]).

Beyond vertical coordination, horizontal collaboration among logistics service providers has also received increasing attention. Horizontal cooperation allows logistics companies to share transportation resources, consolidate shipments, and improve capacity utilization. Previous research shows that cooperation among carriers can generate significant operational and economic benefits (Cruijssen et al., 2007 [3]; Badraoui et al., 2020 [4]). In such collaborative systems, participating firms may share transportation routes, vehicles, or logistics infrastructure to improve system-wide efficiency. Recent platform-oriented work has extended this line by integrating multi-agent systems and digital twins to facilitate carrier collaboration and resource visibility (Xu et al., 2025 [5]).

Another important research direction investigates how the benefits generated by cooperation should be allocated among participants. Cooperative game theory has been used to study cost allocation and profit distribution problems in collaborative logistics systems. Methods such as the Shapley value have been applied to allocate gains from cooperation among firms in a fair manner (Krajewska et al., 2008 [6]; Ferrell et al., 2020 [7]). However, cooperative game theory approaches often require strong assumptions about coalition formation, transferable utility, and complete information, which may be difficult to satisfy in open logistics networks where firms are loosely connected.

With the emergence of new logistics paradigms such as the Physical Internet (PI), coordination among independent firms has become even more important. The PI envisions an open logistics system where standardized containers, shared logistics infrastructure, and digital coordination platforms enable efficient movement of goods through logistics networks (Montreuil, 2011 [8]). In such open systems, coordination mechanisms must consider interactions among multiple independent firms operating in shared logistics environments (Ballot et al., 2014 [9]). At the urban scale, Matusiewicz (2024) [10] proposed a PI deployment framework centered on stakeholder collaboration and shared logistics assets.

2.2. Optimization-Based Logistics Coordination

In addition to contractual and cooperative approaches, some studies investigate logistics coordination using optimization models. Mathematical programming has been applied in logistics network design, transportation planning, and freight system optimization. These models aim to improve system efficiency by minimizing total logistics costs or maximizing overall network performance.

Early studies applied optimization models to freight transportation planning and logistics network design (Crainic & Laporte, 1997 [11]). With increasing computational power and data availability, optimization-based models have become an important tool for analyzing complex logistics systems with multiple transportation modes and capacity constraints.

More recently, optimization approaches have been applied to emerging logistics systems such as Physical Internet networks. For example, Yang et al. (2017) [12] proposed an inventory coordination model based on interconnected logistics services within the Physical Internet framework. Similarly, Sternberg et al. (2021) [13] employed a linear programming model to examine PI-container repositioning in a domestic network. The model minimizes flow imbalances between hubs and is used to analyze how container compatibility influences forward and reverse flows, as well as overall network efficiency. Recent PI optimization studies have examined hub-location and routing decisions under shared warehousing and trucking, as shown by Naganawa et al. [14]. Boysen et al. [15] analyzed split transports in open PI networks and found that, except for split deliveries, their additional operational value is limited.

Recent research has also explored logistics coordination under resource constraints and sustainability considerations. For example, city logistics studies (e.g., Crainic et al. (2023) [16]) have widely applied operations research methods to support planning and management decisions in complex urban freight systems. These approaches address supply planning problems at strategic, tactical, and operational levels, with the objective of improving service quality, economic efficiency, and environmental sustainability. However, most optimization-based logistics coordination models primarily focus on improving system efficiency. The objective functions of these models typically minimize total logistics costs or maximize system performance, while the distribution of costs and benefits among participating firms receives relatively limited attention.

2.3. Fairness-Oriented Optimization

In recent years, fairness considerations have increasingly been incorporated into optimization models for multi-agent systems. Fairness-oriented optimization aims to balance efficiency and equity among multiple decision-makers, particularly in systems where shared resources are allocated among independent participants.

One widely used approach is the concept of min–max fairness, which seeks to minimize the maximum disadvantage experienced by any participant. Such approaches have been studied in resource allocation and communication networks (Radunovic and Le Boudec, 2007 [17]), fairness–efficiency trade-off analysis in operations research (Bertsimas et al., 2011 [18]), transportation task allocation problems (Ye et al., 2017 [19]), and collaborative profitability balancing among firms (Huang and Ohmori, 2021 [20]). More recently, min–max fairness has also been applied in machine learning and multi-agent allocation problems (Harada et al., 2025 [21]). Fairness-based optimization methods help ensure that no participant is excessively disadvantaged compared with others. However, limited studies have examined how min–max fairness interacts with transportation allocation and compensation mechanisms in collaborative logistics systems under shared transportation capacity constraints.

In the context of open Physical Internet logistics systems, min–max fairness is particularly suitable because participating firms are often loosely connected and may not form stable cooperative coalitions. Unlike Shapley-based allocation methods, which require assumptions regarding transferable utility and marginal contribution estimation, the min–max approach directly controls the worst-off participant under shared resource constraints. Moreover, compared with proportional fairness or envy-freeness concepts, min–max fairness provides a more operationally interpretable criterion for maintaining participation stability in logistics coordination problems, because it directly limits the worst disadvantage among participating firms. This allows coordination platforms or policymakers to implement practical rules such as preventing any single firm from bearing excessive additional logistics burden under shared capacity constraints. In contrast, proportional fairness focuses on relative benefit ratios, which may be difficult to interpret among firms with different operational scales, while envy-freeness is less straightforward to define in logistics systems where firms have heterogeneous route preferences and delivery priorities.

Fairness considerations have also been studied in resource allocation and operational decision-making problems. For example, recent studies investigate fairness-aware optimization models that balance system efficiency with equitable outcomes among decision makers (Hooker & Williams, 2012 [22]; Karsu & Morton, 2015 [23]; Babaioff et al., 2024 [24]). In collaborative freight settings, Lai et al. (2025) [25] jointly optimized multi-stop truckload routes and least-core cost allocations, whereas Gückel et al. (2025) [26] incorporated cost- and workload-based fairness constraints directly into multi-day city logistics planning.

2.4. Gaps

Despite these advances, fairness considerations have not been sufficiently integrated into coordinated logistics planning under shared transportation capacity constraints. Existing studies mainly focus on improving system efficiency, while the distribution of benefits among participating firms is often overlooked.

Moreover, even when compensation mechanisms are introduced, fairness is typically achieved through financial transfers rather than adjustments in resource allocation. As summarized in

Table 1. Positioning of this study relative to representative fairness and allocation studies.

Study Category	Fairness Concept	Application Domain	Compensation Mechanism	Shared Transportation Capacity
Collaborative logistics allocation studies [6]	Request allocation and profit sharing	Horizontal freight carrier cooperation	Yes, through profit-sharing allocation	Not the main focus
Communication network fairness studies [17]	Max–min fairness	Communication networks/bandwidth allocation	No	Shared resource capacity
OR fairness trade-off studies [18]	Various fairness measures	General optimization problems	No explicit mechanism	No
Transportation allocation studies [19]	Weighted/normalized fairness	Transportation systems	No explicit mechanism	Yes (task/resource sharing)
Collaborative profitability studies [20]	Fairness-oriented pricing	Supply chain coordination	compensation-like transfer pricing	No
Collaborative cost-allocation study [25]	Least-core cost allocation	Collaborative multi-stop truckload shipping	Yes, through ex-post cost allocation	Vehicle capacity is embedded in routing
Cooperative city logistics study [26]	Cost- and workload-based fairness constraints	Cooperative two-tier city logistics	No explicit compensation budget	Shared services and resources; route capacity is not the main focus
This study	Min–max disadvantage balancing	Collaborative logistics systems/Physical Internet	Yes, limited compensation budget	Yes, shared route capacity

, many previous studies have examined fair cost/profit allocation or resource allocation in collaborative logistics. However, relatively few studies explicitly examine how limited compensation budgets interact with route-allocation adjustments and shared transportation capacity constraints in a PI-oriented coordination setting. As a result, it remains unclear whether fairness can be achieved through minor changes in logistics decisions without significantly altering the overall system structure.

In addition, the role of system conditions in shaping fairness outcomes has not been sufficiently explored. In many practical coordination settings, compensation budgets may be limited, and shared transportation capacity may be constrained. However, how these factors jointly influence the effectiveness of fairness-oriented coordination remains underexplored.

To address these gaps, this study develops an optimization framework for coordinated logistics planning in open logistics networks (Physical Internet systems). The proposed framework integrates efficiency-oriented coordination with a fairness-oriented min–max model, enabling the analysis of fairness improvement through allocation adjustments under a limited compensation budget and shared capacity constraints.

3. Model Formulation

3.1. Problem Setting

A coordinated logistics planning problem is considered in which multiple firms share a limited set of transportation routes between a common origin and destination. Each firm has a given shipment demand that must be transported through the available routes. Each route is characterized by transportation cost, delivery time, carbon emissions, and limited capacity.

The decision problem is to determine how to allocate the shipment of each firm across the available routes. In addition to route allocation, compensation may be provided to firms to ensure participation in the coordinated system. A limited compensation budget is assumed.

The system is subject to several constraints, including demand satisfaction for each firm, capacity limits of each route, and the total compensation budget.

Two optimization models are developed as two alternative coordination rules. They should not be understood as a simple improvement from Model 3.3 to Model 3.4. The first rule, Model 3.3, minimizes total system cost while imposing strict individual rationality. It guarantees that no firm becomes worse off after compensation. This rule is suitable for the initial or pilot stage of collaboration, when firms have limited trust in the platform and need a no-loss guarantee to participate. The second rule, Model 3.4, uses a fairness-oriented min–max objective. It minimizes the maximum disadvantage among firms. This rule is suitable for mature or repeated coordination settings, especially when compensation resources are limited. In this stage, the platform may not be able to guarantee no loss in every single period. Instead, it tries to prevent any firm from facing excessive or repeated disadvantage over time. The coordination perspectives of the two rules are summarized in Figure 1.

3.2. Sets, Parameters, and Decision Variables

Sets

$I$ set of firms (players);

$R$ set of transportation routes.

Parameters

$d_i$ shipment demand of firm $i$ (container units);

$c_r$ capacity of route $r$ (container units);

${\gamma }_{ir}$ transportation cost per unit on route $r$ from firm $i$ (kJPY/unit);

${\eta }_{ir}$ CO₂ emission index of firm $i$ on route $r$ (index/unit);

${\tau }_{ir}$ transportation time of route $r$ from firm $i$ (h);

${\overline{\tau }}_i$ expected delivery time of firm $i$ (h);

${\delta }_{ir}$ positive delay of firm $i$ on route $r$ (h);

$p_{{CO}_2}$ unit cost of CO₂ emission index (kJPY/index);

$p_{delay}$ delay penalty coefficient (kJPY/h);

${\theta }_i^{solo}$ minimum logistics cost of firm $i$ under standalone operation;

$B$ total compensation budget (kJPY).

Decision variables

$x_{ir}$ shipment quantity of firm $i$ assigned to route $r$;

$z_i$ compensation allocated to firm $i$;

$\lambda$ maximum disadvantage among firms in the fairness-oriented model.

3.3. Efficiency-Oriented Coordination Model

The logistics cost of firm $i$ under coordinated planning is defined as

$${\theta }_i(x_i)={\sum }_{r\in R}{x}_{ir}({\gamma }_{ir}+p_{{CO}_2}{\eta }_{ir}+p_{delay}{\delta }_{ir})$$

(1)

where the term $p_{{CO}_2}{\eta }_{ir}$ represents the carbon emission index, and $p_{delay}{\delta }_{ir}$ represents the penalty cost associated with delivery delay.

The delay parameter represents the positive delay of firm $i$ on route $r$, which is precomputed from the given route travel time and expected delivery time:

$${\delta }_{ir}=\mathrm{m}\mathrm{a}\mathrm{x}(0,{\tau }_{ir}-{\overline{\tau }}_i)$$

(2)

The efficiency-oriented coordination model aims to minimize the total system costs; the efficiency-oriented coordination model is

$$\min \sum _{i\in I}{\theta }_i\left(x_i\right)$$

(3)

subject to

Demand satisfaction

$$\sum _{r\in R}{x}_{ir}=d_i,\forall i\in I$$

(4)

Route capacity constraint

$$\sum _{i\in I}{x}_{ir}\le c_r,\forall r\in R$$

(5)

Participation constraint

$${\theta }_i\left(x_i\right)-z_i\le {\theta }_i^{solo},\forall i\in I$$

(6)

Compensation budget

$$\sum _{i\in I}{z}_i\le B$$

(7)

Non-negativity constraints

$$x_{ir}\ge 0, \forall i,r$$

(8)

$$z_i\ge 0, \forall i$$

(9)

The objective function (3) minimizes the total cost of all firms, representing an efficiency-oriented coordination approach that focuses on system-wide performance.

Constraint (4) ensures that the shipment demand of each firm is satisfied. Constraint (5) enforces the capacity limitation of each route. Constraint (6) guarantees that no firm becomes worse off than its standalone operation after receiving compensation. Constraint (7) restricts the total compensation within the available budget. This model focuses on improving overall system efficiency through coordinated logistics planning.

3.4. Fairness-Oriented Min–Max Model

When the compensation budget is limited, it may not be possible to fully offset the differences in outcomes among firms through compensation alone. In such cases, fairness considerations become important.

Unlike Model 3.3, which guarantees individual rationality, Model 3.4 minimizes the maximum disadvantage among firms under a limited compensation budget. From a participation perspective, Model 3.4 should be understood as a rule for a mature or repeated coordination system. It is not a one-period contract that every firm will always accept. In one period, some firms may become worse off than in the standalone benchmark. However, the min–max objective limits the worst disadvantage. In other words, it prevents the most disadvantaged firm from bearing too large a burden. In repeated coordination, temporary losses can be reduced later through future compensation, priority access to shared capacity, or rotation rules. Therefore, the role of Model 3.4 is not to guarantee that every firm is better off in every period. Its role is to prevent excessive or repeated disadvantage when the compensation budget is limited.

The disadvantage of firm $i$ is defined as

$${\Delta }_i={\theta }_i\left(x_i\right)-z_i-{\theta }_i^{solo}$$

(10)

This value represents the cost increase relative to standalone operation.

To reduce inequality among firms, the following min–max model is introduced:

$$\min \lambda$$

(11)

subject to

$${\theta }_i\left(x_i\right)-z_i-{\theta }_i^{solo}\le \lambda , \forall i\in I$$

(12)

Demand satisfaction

$$\sum _{r\in R}{x}_{ir}=d_i, \forall i\in I$$

(4′)

Route capacity constraint

$$\sum _{i\in I}{x}_{ir}\le c_r, \forall r\in R$$

(5′)

Compensation budget

$$\sum _{i\in I}{z}_i\le B$$

(7′)

Non-negativity constraints

$$x_{ir}\ge 0, \forall i,r$$

(8′)

$$z_i\ge 0, \forall i$$

(9′)

The objective of the model is to minimize the variable $\lambda$, which represents the maximum disadvantage among all firms.

Constraint (12) ensures that, for each firm $i$, the difference between its cost under coordination ${\theta }_i\left(x_i\right)$, adjusted by compensation $z_i$, and its standalone cost ${\theta }_i^{solo}$, does not exceed $\lambda$. By minimizing $\lambda$, the model seeks to reduce the worst-case disadvantage and thus improve fairness among firms.

Compared with the efficiency-oriented coordination model, which focuses on improving total system efficiency, the min–max model explicitly addresses fairness among participants when compensation resources are limited.

Together, these two models enable the analysis of the efficiency–fairness trade-off in coordinated logistics planning.

4. Numerical Experiments

4.1. Experimental Setting

4.1.1. Target Network

A Japan-inspired domestic logistics network is generated for consumer goods distribution. Firms represent consumer goods producers, processors, or logistics service providers located around major metropolitan and industrial regions, while destinations represent regional consumer markets such as Hokkaido, Kansai, and Kyushu. The network includes major logistics regions such as Tokyo, Sendai, Nagoya, Osaka, Hiroshima, Fukuoka, and Sapporo.

Figure 2 illustrates the logistics network. Route alternatives are defined for each origin–destination corridor and include truck, rail, coastal shipping, and ferry/intermodal services. A firm can use only the route alternatives that correspond to its own origin and destination, which avoids assigning shipments to unrelated corridors and provides a more realistic representation of domestic freight distribution than a single common origin–destination setting.

4.1.2. Benchmark Scenario and Compared Models

The decentralized benchmark is computed using sequential allocation, which represents an uncoordinated first-come, first-served use of shared route capacity. In this benchmark, firms are ordered exogenously, assigned transportation resources one by one, and each firm selects the currently available route alternatives with the lowest unit logistics cost given the remaining capacities. After each firm’s allocation is fixed, the residual capacity is updated and becomes the feasible capacity set for the next firm. Because the outcome of a sequential process may depend on the order in which firms access capacity, the benchmark cost is obtained by averaging over multiple allocation orders. This benchmark is suitable for comparison because it captures a practical decentralized situation in which firms make individually rational routing decisions without a system-level coordinator while still facing the same route-capacity constraints as the coordinated models.

Two coordination rules are compared with the benchmark. Model 3.3 is the efficiency-oriented participation-guaranteeing rule. It minimizes the total coordinated logistics cost and ensures that no firm becomes worse off than in the benchmark after compensation. Model 3.4 is the fairness-oriented rule. It minimizes the maximum firm-level disadvantage under the compensation budget. The main evaluation indicators are total cost reduction and maximum disadvantage.

4.1.3. Parameter Settings

Because the instances are generated from multiple corridors rather than from a single fixed data table, parameter ranges are reported instead of listing every firm and route. The original four-firm base case is introduced here as the reference setting: all firms share one origin–destination pair, use three route alternatives, and have a compensation budget of 5000 kJPY.

Table 2. Firm-specific parameter settings.

Parameter	Base Case	Large-Scale Instances
Number of firms	4	8, 12, 20, 30, 50
Origin–destination corridors	1 common OD pair	Up to 7 Japan-inspired OD corridors
Demand per firm	60–90 container units	35–105 container units
Expected delivery time	14–22 h	10.0–50.8 h

summarizes firm-specific settings, while

Table 3. Route-specific parameter settings.

Parameter	Base Case	Large-Scale Instances
Route cost	600–800 kJPY/unit	518.4–1273.9 kJPY/unit
Travel time	12–22 h	8.6–58.4 h
CO2 emission index	6–20 index/unit	4.6–33.5 index/unit
Route capacity	90–180 container units	64.5–848.3 container units
Route availability	All firms share r1–r3	Only OD-matching routes are available to each firm

summarizes route-specific settings.

4.2. Results

4.2.1. Results Across Different Problem Sizes

Table 4. Computational results for network instances.

Firms	Avg. Active OD	Avg. Routes	Benchmark Cost	M3.3 Cost Red. (%)	M3.4 Cost Red. (%)	Wall Time (s)
4	1.0	3.0	566,800	1.12	0.27	0.0010
8	5.2	15.6	841,562	0.16	0.02	0.0011
12	5.8	17.4	1,075,528	0.52	0.10	0.0014
20	6.2	18.6	1,908,953	4.68	0.27	0.0018
30	6.8	20.4	2,770,949	2.40	0.18	0.0022
50	7.0	21.0	4,341,069	1.13	0.13	0.0031

reports the average computational results. The four-firm row corresponds to the original base experiment, whereas the larger rows report the average of five generated instances for each size. The reported time is wall-clock time for model construction and optimization through the Gurobi Python interface. The computations were performed using Python 3.12.4 and Gurobi Optimizer 11.0.3.

The results show that the coordinated models consistently improve upon the decentralized benchmark. Model 3.3 generally achieves a larger cost reduction because it directly minimizes total system cost while maintaining individual rationality. In contrast, Model 3.4 produces smaller cost reductions because it prioritizes reducing the worst firm-level disadvantage.

The computational times remain small for all tested instances because the models are continuous linear programs. This means that the framework can be used for the Japan-inspired network sizes tested in this study. However, fast computation is not the main contribution. It only shows that the framework is practical to use. The main insight is that different coordination rules produce different benefit distributions. This helps a platform choose a rule according to its development stage, participant trust, and available compensation budget.

4.2.2. Detailed Analysis of the Base Case

The base case is retained to provide an interpretable explanation of how the two coordination models differ at the firm and route-allocation levels. The results are analyzed from three perspectives: firm-level fairness, system-level efficiency, and allocation adjustment mechanisms, as illustrated in Figure 3, Figure 4 and Figure 5.

(1)

Firm-level fairness outcomes

Under Model 3.3, the disadvantage values are unevenly distributed across firms. While firms A, B, and D remain at the benchmark level (zero disadvantage), firm C achieves a substantial improvement with a reduction of 11,349 kJPY. This occurs because the individual rationality constraints become binding under the limited compensation budget. Since the objective of Model 3.3 is to minimize total coordinated cost, the optimization primarily allocates efficiency gains to firm C through route reallocation, while ensuring that the remaining firms do not become worse off than the decentralized benchmark. These results indicate that the efficiency-oriented model concentrates benefits on a subset of firms while maintaining individual rationality.

In contrast, Model 3.4 produces identical disadvantage values (−1638 kJPY) for all firms in the base case. This demonstrates that the min–max formulation successfully equalizes outcomes across firms in this scenario, achieving a balanced allocation of benefits.

(2)

System-level efficiency

Both coordination models reduce the total cost compared with the benchmark (566,800 kJPY), confirming the benefit of collaboration. Model 3.3 achieves the lowest total cost (560,451 kJPY), reflecting its efficiency-oriented objective.

However, Model 3.4 results in a higher total cost (565,247 kJPY) than Model 3.3. This increase represents the cost of achieving fairness, indicating a clear trade-off between efficiency and equity in coordinated logistics planning.

(3)

Allocation adjustment mechanism

The results show that adjustments are relatively small and occur mainly as reallocations between routes r1 and r2 within individual firms. For example, firm C shifts part of its allocation from r2 to r1, while firm D exhibits the opposite pattern.

Notably, the allocation on route r3 remains unchanged across all models, indicating that this route is effectively dedicated to firm A due to its compatibility with delivery time constraints. This suggests that structural constraints limit the extent of feasible reallocation.

Despite the small magnitude of these adjustments, their impact on fairness is significant. Model 3.4 achieves equalized outcomes across firms through subtle redistribution rather than large-scale changes in allocation.

(4)

Relationship between allocation adjustment and fairness

The results show that fairness improvements can be achieved through relatively small allocation adjustments. Interestingly, both Model 3.3 and Model 3.4 use the same total compensation budget. Therefore, the observed differences are mainly caused by changes in transportation allocation and route assignment rather than by additional compensation itself. This suggests that fairness improvements can be achieved through structural coordination of logistics operations even under limited compensation resources.

As shown in Figure 5, the changes in transport allocation relative to the benchmark are quantitatively small. Most adjustments occur as marginal shifts between routes r1 and r2 within individual firms, typically on the order of only a few units. Moreover, route r3 remains completely unchanged across all scenarios, indicating that the overall structure of the logistics network is largely preserved.

Despite these limited changes, the impact on fairness is substantial. Under Model 3.3, the benefits of coordination are concentrated in a single firm (firm C), while the remaining firms experience no improvement. In contrast, Model 3.4 redistributes these benefits evenly across all firms, achieving identical disadvantage levels.

This contrast highlights that fairness improvements do not necessarily require large-scale reconfiguration of the system. Instead, relatively small reallocations, when combined with an appropriate compensation mechanism, can alter the distribution of outcomes.

From a practical perspective, this is particularly important. Large structural changes in logistics networks are often costly and difficult to implement, whereas marginal adjustments are more feasible in real-world operations. The results therefore suggest that fairness-oriented coordination can be achieved with minimal disruption to existing logistics structures.

4.3. Sensitivity Analysis

4.3.1. Impact of Compensation Budget

To further investigate the impact of the compensation mechanism, a sensitivity analysis is conducted with respect to the total compensation budget. The budget varies from 0 to 12,000 kJPY, and the system performance is evaluated under both Model 3.3 (efficiency-oriented with IR) and Model 3.4 (fairness-oriented min–max model). The corresponding effect on system efficiency is shown in Figure 6.

(1)

Impact on System Efficiency

A clear difference is observed between the two models. Under Model 3.3, the cost reduction increases rapidly with the budget, exhibiting a steep and nearly linear trend. This indicates that the compensation budget is effectively utilized to enable more efficient routing decisions. In particular, the total cost reduction reaches 15,238 kJPY at a budget of 12,000 kJPY, outperforming Model 3.4.

In contrast, Model 3.4 shows only limited efficiency improvement. Even at the highest budget level, the cost reduction is only 3726 kJPY. This difference arises from the distinct role of compensation in the two models. In Model 3.3, compensation relaxes the individual rationality (IR) constraints, allowing the system to achieve globally optimal allocations. Therefore, each additional unit of budget contributes not only to compensation but also to system-wide efficiency gains.

On the other hand, Model 3.4 allocates the budget primarily to reduce inequality among firms rather than to improve routing efficiency. As a result, the marginal impact of budget on cost reduction remains relatively small.

(2)

Impact on Fairness

Under Model 3.3, the maximum disadvantage remains at zero for all budget levels. This indicates that the IR constraint is always binding, meaning that no firm is worse off compared to the benchmark. The system ensures participation feasibility but does not generate surplus benefits for individual firms.

In contrast, Model 3.4 exhibits a continuous decrease in the maximum disadvantage as the budget increases. The disadvantage becomes negative even at small budget levels and reaches −3932 kJPY at the highest budget. In the tested budget scenarios, all firms become better off than in the decentralized benchmark. However, this result is specific to these numerical scenarios. It should not be understood as a general guarantee of Model 3.4. This is because Model 3.4 minimizes the worst disadvantage but does not impose individual rationality constraints for all firms. These fairness results are shown in Figure 7.

Moreover, the relationship between the budget and fairness improvement appears approximately linear over the tested range. This means that the compensation budget can be used as a practical tool for setting a fairness target. For example, the platform can decide how much worst-firm disadvantage is acceptable and then consider the required compensation budget. Therefore, the budget should be evaluated not only by total cost reduction but also by how much it improves the worst firm-level outcome.

4.3.2. Impact of Shared Route Capacity Reduction

To examine the robustness of the coordination model under resource scarcity, the capacities of the shared routes (rail r2 and ship r3) are progressively reduced from 100% to 75% of their original levels, while the capacity of the truck route remains unchanged. This setting uses a higher rail cost, 1400 kJPY/unit, to reflect increasing competition for commonly preferred transport options in shared logistics systems.

(1)

Impact on System Efficiency

Figure 8 shows the effect of shared-route capacity reduction on total logistics cost. When the capacity scale decreases from 1.00 to 0.90, both the benchmark cost and the coordinated cost remain unchanged, indicating that the system is robust to moderate capacity reductions. In this range, the available capacity is still sufficient to accommodate demand without significantly altering route allocation.

However, once the capacity scale falls below approximately 0.85, both costs begin to increase noticeably. This suggests that capacity constraints become binding and force part of the flow onto less preferred routes, leading to a rapid deterioration in system efficiency.

Despite this deterioration, the coordinated solution under Model 3.4 consistently maintains a lower total cost than the benchmark across all scenarios. Moreover, the coordination advantage widens as capacity becomes scarce. These results demonstrate that coordination continues to generate efficiency gains even when the system becomes highly constrained.

(2)

Impact on Fairness

Figure 9 illustrates the variation of the maximum disadvantage λ under different capacity levels. In the tested capacity-reduction scenarios, λ remains negative. This means that all firms are better off than in the benchmark in these numerical cases. Therefore, the results show a Pareto-improving outcome for the tested instances. However, Model 3.4 does not guarantee Pareto improvement in general because it does not include individual rationality constraints.

A notable observation is that λ becomes more negative as the capacity decreases, improving from −1533 to approximately −3533. This implies that all firms benefit more from coordination when the system becomes more constrained. The reason is that the benchmark solution deteriorates more rapidly under capacity scarcity, whereas the coordinated model mitigates this effect through more balanced allocation and compensation.

However, this improvement is not unlimited. When the capacity scale falls below approximately 0.80, λ stabilizes and no longer decreases. This plateau indicates that fairness improvement is ultimately constrained by the fixed compensation budget. Since the available budget is fully utilized, further capacity reduction cannot improve the worst-off outcome.

4.3.3. Integrated Insights from Budget and Capacity Sensitivity

The two sensitivity analyses show that compensation budget and shared-route capacity have different roles. The compensation budget mainly affects how benefits are distributed among firms. As shown in Section 4.3.1, increasing the budget improves fairness under Model 3.4. In practical terms, the budget can be used to set a fairness target, such as an acceptable upper bound on the worst firm-level disadvantage.

In contrast, shared-route capacity affects the structural conditions of the system. As shown in Section 4.3.2, capacity reduction intensifies competition and leads to threshold effects in system efficiency. At the same time, the relative advantage of coordination becomes more pronounced under capacity scarcity.

Fairness improvement is limited by both the compensation budget and shared-route capacity. Increasing the budget can improve fairness over the tested range. However, the capacity analysis shows that fairness gains stop increasing when the budget is fully used. This means that compensation alone cannot solve all fairness problems if shared capacity is too scarce. Capacity planning and compensation planning should therefore be considered together.

Overall, effective coordination requires both sufficient compensation resources and sufficient shared capacity. For platform operators or policymakers, fairness targets should be set together with the compensation budget and realistic route-capacity conditions. These targets should also depend on the stage of the coordination platform. A no-loss rule may be better in the early stage, while a fairness-balancing rule may be better in a mature or repeated coordination stage.

5. Discussion

5.1. Implications for Theory

This study extends coordinated logistics planning theory by treating fairness not only as a participation requirement but also as a central optimization objective under shared capacity constraints. In conventional efficiency-oriented coordination models, fairness is often represented by an individual rationality constraint, which ensures that no firm becomes worse off than in the benchmark case. In contrast, the proposed min–max formulation explicitly minimizes the maximum firm-level disadvantage. This shift clarifies how fairness can be modeled as an active design objective in Physical Internet-oriented logistics systems where multiple firms share limited transport capacity.

The results also clarify that fairness does not necessarily require large changes in the physical allocation of transport flows. In the numerical experiments, relatively small shifts between routes substantially changed the distribution of firm-level outcomes. This finding suggests that fairness in collaborative logistics is determined not only by the scale of operational adjustment but also by the way coordination outcomes are evaluated and distributed among participants.

Another theoretical implication is that the compensation budget and shared-route capacity act as boundary conditions for fair coordination. A larger compensation budget improves the ability to reduce the maximum disadvantage, while limited shared capacity constrains the feasible set of fair allocations. Therefore, fairness should be understood as an outcome jointly shaped by financial compensation and physical network capacity.

5.2. Implications for Practice and Policy

For logistics managers and platform operators, the proposed framework can be used as a decision-support tool for comparing alternative coordination rules. The efficiency-oriented model is useful when the main objective is to reduce total logistics cost, whereas the fairness-oriented model is more suitable when stable participation by multiple firms is required. By explicitly measuring the maximum disadvantage, the model helps identify whether any participant is likely to be worse off under a proposed coordination plan.

The results indicate that a compensation mechanism is important for practical implementation. Compensation can make coordination more acceptable by redistributing part of the system-wide benefit to firms that would otherwise receive limited gains. Because the budget level directly affects the fairness outcome, platform operators can use the compensation budget as a practical policy lever when setting fairness targets among participating firms.

The capacity analysis also provides useful guidance for infrastructure and policy design. When shared rail or maritime capacity becomes tight, coordination can still reduce total cost and improve fairness, but the improvement eventually reaches a limit. This means that financial compensation alone is not sufficient. Policies that maintain or expand shared transport capacity, improve access to intermodal routes, and support transparent coordination rules are also needed for sustainable collaboration.

5.3. Limitations of the Study and Future Research Directions

This study has several limitations. First, the numerical experiments are based on a stylized intercity logistics network with deterministic demand, cost, and capacity settings. Although this design makes the mechanism clear, future research should test the model with larger empirical networks, time-varying demand, uncertain travel conditions, and disruption scenarios.

Second, the model assumes that a central coordinator can collect the necessary information and determine allocations and compensation. In real logistics systems, firms may have private information, different bargaining power, and strategic incentives. Future studies should examine decentralized coordination, negotiation processes, incentive-compatible mechanisms, and privacy-preserving data sharing.

6. Conclusions

This study developed an optimization framework for coordinated logistics planning with shared transportation capacity. The framework includes two complementary coordination rules. Model 3.3 is an efficiency-oriented participation-guaranteeing rule with individual rationality constraints. Model 3.4 is a fairness-oriented rule that minimizes the maximum disadvantage among firms under a limited compensation budget. The originality of the framework is that these two rules are connected with route allocation, compensation limits, shared capacity constraints, and the development stage of a PI logistics platform.

The numerical results reveal several important findings. First, coordinated planning reduces total logistics cost compared with decentralized allocation, confirming the value of resource sharing. The efficiency-oriented model achieves the lowest system cost but leads to uneven benefit distribution, with gains concentrated in a subset of firms. In contrast, the fairness-oriented model produces equalized outcomes across all firms, demonstrating that fairness-oriented coordination can effectively balance benefits among participants under shared transportation capacity constraints.

Second, fairness improvements can be achieved through relatively small adjustments in route allocation. The results show that only minor shifts between routes are sufficient to change the distribution of outcomes, while the overall structure of the logistics network remains largely unchanged. This suggests that fairness-oriented coordination can be implemented with limited operational disruption and does not necessarily require large-scale restructuring of transportation operations.

Third, the sensitivity analyses show the different roles of compensation budget and shared-route capacity. The compensation budget mainly controls the fairness level. The results also show that fairness improvements are mainly caused by changes in transportation allocation and route assignment, not only by compensation itself. Shared-route capacity affects the structure of the system and creates threshold effects in system efficiency. When capacity becomes limited, coordination becomes more important because it reduces the loss caused by decentralized decisions. However, fairness improvement stops when the compensation budget is fully used. These findings suggest that managers should choose fairness targets together with compensation budgets, capacity policies, and the development stage of the platform. Model 3.3 may be better in the early stage. Model 3.4 may be more useful in mature or repeated coordination settings when the platform wants to avoid excessive or repeated disadvantage.

Overall, the findings provide a basis for designing fair and efficient coordination mechanisms for shared-capacity logistics systems.