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The Multi-Commodity Flow Problem with Outsourcing Decisions

Elena Fernandez, Ivana Ljubic, Nicolas Zerega

TL;DR

The paper studies a novel prize-collecting, Stackelberg routing problem called the Multi-Commodity Flow Problem with Outsourcing Decisions (MCFOD), where a leader may outsource the first and last legs of commodity shipments to external carriers and selects which commodities to serve. The authors develop a bilevel Mixed-Integer Nonlinear Program (MINLP) that captures leader decisions, follower responses, and routing costs, then derive single-level reformulations, including several MILP variants such as Explicit Paths (EP) and Implicit Paths (IP). They prove NP-hardness for all variants and show that the follower responses induce discretization of outsourcing fees, enabling tractable reformulations; they also adapt these formulations to fixed-fee settings and relaxed follower rules ($r$ variants). Through extensive computational experiments on CAB and AP benchmark sets, the implicit-path formulation (IP) consistently delivers the best performance, solving instances with up to 200 nodes and 6 carriers within practical time, and yielding managerial insights about outsourcing fees, follower behavior, and service levels. The study demonstrates substantial profit and service-rate improvements when the leader optimizes outsourcing fees and allocations, and highlights the impact of follower strategies on overall outcomes, with implications for freight and hub-based logistics networks.

Abstract

We address a new prize-collecting problem of routing commodities in a given network with hub and non-hub nodes, in which the service of the non-hub nodes will be outsourced to third-party carriers. The problem is modeled as a Stackelberg game: there is a major firm (leader) that decides to serve a subset of commodities. The leader aims to outsource first and third legs of transportation services to smaller carriers (who act as followers) by allocating at most one carrier to each non-hub node. The carriers try to maximize their own profits, which are influenced by the leader's offers. The goal of the leader is to determine the optimal outsourcing fees, along with the allocation of carriers to the non-hub nodes, so that the profit from the routed commodities is maximized. The optimal response of the followers must be taken into account, as the followers might refuse to serve some legs in case they are negative or do not maximize their profit. We also study two alternative settings: one in which the outsourcing fees are fixed, and the other one in which the carriers accept any offer, as long as the resulting profit is non-negative. We prove that the set of possible outsourcing fees can be discretized and formulate the problem as a single-level mixed-integer nonlinear program. For all considered problem variants, we prove NP-hardness and propose and computationally investigate several MIP formulations. We study the computational scalability of these MIP formulations and analyze solutions obtained by varying the reservation prices of the carriers. Finally, by comparing the introduced problem variants, we derive some interesting managerial insights.

The Multi-Commodity Flow Problem with Outsourcing Decisions

TL;DR

The paper studies a novel prize-collecting, Stackelberg routing problem called the Multi-Commodity Flow Problem with Outsourcing Decisions (MCFOD), where a leader may outsource the first and last legs of commodity shipments to external carriers and selects which commodities to serve. The authors develop a bilevel Mixed-Integer Nonlinear Program (MINLP) that captures leader decisions, follower responses, and routing costs, then derive single-level reformulations, including several MILP variants such as Explicit Paths (EP) and Implicit Paths (IP). They prove NP-hardness for all variants and show that the follower responses induce discretization of outsourcing fees, enabling tractable reformulations; they also adapt these formulations to fixed-fee settings and relaxed follower rules ( variants). Through extensive computational experiments on CAB and AP benchmark sets, the implicit-path formulation (IP) consistently delivers the best performance, solving instances with up to 200 nodes and 6 carriers within practical time, and yielding managerial insights about outsourcing fees, follower behavior, and service levels. The study demonstrates substantial profit and service-rate improvements when the leader optimizes outsourcing fees and allocations, and highlights the impact of follower strategies on overall outcomes, with implications for freight and hub-based logistics networks.

Abstract

We address a new prize-collecting problem of routing commodities in a given network with hub and non-hub nodes, in which the service of the non-hub nodes will be outsourced to third-party carriers. The problem is modeled as a Stackelberg game: there is a major firm (leader) that decides to serve a subset of commodities. The leader aims to outsource first and third legs of transportation services to smaller carriers (who act as followers) by allocating at most one carrier to each non-hub node. The carriers try to maximize their own profits, which are influenced by the leader's offers. The goal of the leader is to determine the optimal outsourcing fees, along with the allocation of carriers to the non-hub nodes, so that the profit from the routed commodities is maximized. The optimal response of the followers must be taken into account, as the followers might refuse to serve some legs in case they are negative or do not maximize their profit. We also study two alternative settings: one in which the outsourcing fees are fixed, and the other one in which the carriers accept any offer, as long as the resulting profit is non-negative. We prove that the set of possible outsourcing fees can be discretized and formulate the problem as a single-level mixed-integer nonlinear program. For all considered problem variants, we prove NP-hardness and propose and computationally investigate several MIP formulations. We study the computational scalability of these MIP formulations and analyze solutions obtained by varying the reservation prices of the carriers. Finally, by comparing the introduced problem variants, we derive some interesting managerial insights.
Paper Structure (44 sections, 6 theorems, 41 equations, 3 figures, 18 tables)

This paper contains 44 sections, 6 theorems, 41 equations, 3 figures, 18 tables.

Table of Contents

  1. Introduction
  2. Our Contribution:
  3. Literature Overview.
  4. Problem definition
  5. Input Data:
  6. Decisions of the leader:
  7. Decisions of the Followers:
  8. Bilevel Formulation and a Single-Level MINLP Reformulation
  9. Bilevel MILP formulation for the MCFOD
  10. The carriers' subproblems.
  11. Single-level MI(N)LP Reformulation
  12. Solution Properties and Problem Complexity
  13. Optimal outsourcing fees for given non-hub allocations, and first and third legs
  14. Analysis for $r$MCFOD
  15. Problem complexity
  16. ...and 29 more sections

Key Result

Proposition 4.1

Let $\overline f$, $\overline t$, and $\overline a$ be given first and third legs, and non-hub allocations satisfying const:allocate-fixed-const:dr_allocation_MINLP. Then, the outsourcing fees $\overline p$ and $\overline q$ obtained according to best_p and best_q are optimal for $\overline f$, $\ov

Figures (3)

  • Figure 1: Network for Example \ref{['example:1']} with set of commodities $\{(1, 5), (2, 5), (3, 5), (4, 5)\}$.
  • Figure 2: Comparison between optimal MCFOD and MCFOD$_F$ solutions on served commodities and commodity evolution for instances from the CAB dataset.
  • Figure 3: Average total profit comparison for CAB dataset

Theorems & Definitions (8)

  • Definition 2.3: MCFOD and $r$MCFOD with Fixed Outsourcing Fees, MCFOD$_F$ and $r$MCFOD$_F$
  • Example 1
  • Proposition 4.1
  • Proposition 4.2: MCFOD with preprocessed routing costs
  • Proposition 4.3: MCFOD$_F$ with preprocessed routing costs
  • Proposition 4.4
  • Proposition 4.5
  • Proposition 4.6