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A multi-agent and auction-based framework and approach for carrier collaboration

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Logistics Research

Abstract

Carrier collaboration in transportation means multiple carriers form an alliance to optimize their transportation operations through sharing transportation requests and vehicle capacities. In this paper, we propose a multi-agent and auction-based framework and approach for carrier collaboration in less than truckload transportation. In this framework, the carriers outsource/acquire requests through multiple auctions, one for outsourcing each request; a carrier acts as an auctioneer when it wants to outsource a request to other carriers, whereas the carrier acts as a bidder when it wants to acquire a request from other carriers; for each carrier, which requests it should outsource and acquire are determined by solving its outsourcing requests selection problem and requests bidding problem, respectively. These two decision problems are formulated as mixed integer programming problems. The auction of each request is multiround; in each round, the auctioneer determines the outsourcing price of the request and each bidder determines whether it acquires the request at the given price; the auctioneer lowers the outsourcing price if multiple carriers bid for the request or raises the price if no carrier bids for it. The auction process continues until only one carrier bids for the request or a given number of rounds are achieved. In the second case, if no agent bids for the request, then it is returned to the outsourcing agent; if multiple bidding agents compete for the request, a conflict resolution procedure is used to determine which carrier wins it. The approach is decentralized, asynchronous, and dynamic, where multiple auctions may occur simultaneously and interact with each other. The performance of the approach is evaluated by randomly generated instances and compared with an individual planning approach and a centralized planning approach.

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Correspondence to Haoxun Chen.

Appendix: centralized framework for carrier collaboration

Appendix: centralized framework for carrier collaboration

Under the centralized framework, multiple carriers come to an agreement for cooperating each other and constitute a collaborative alliance with a coordinator in charge of making collaborative transportation plans for the alliance. The coordinator may be a virtual one. The objective of the coordinator is to maximize the total profit of the alliance, which will then be allocated among the carriers. Regarding the problem investigated in this paper, the coordinator in the centralized framework determines the reallocation of all transportation requests to the carriers so as to maximize the total profit. This centralized requests reallocation problem (CRRP) for CCPLTL can be formulated as a mixed integer programming (MIP). The notations used in the MIP model include the notations introduced in Sect. 4 and the following notations.

k::

carrier index, k = 1, …,K, where K represents the number of carriers

l::

request index, l = 1, …,L, where L represents the total number of requests of all carriers

C k :

vehicle capacity of carrier k

W k :

the number of vehicles owned by carrier k

o k :

the depot of carrier k

α k :

the minimum profit margin of carrier k

\( c_{ij}^{k} \) :

transportation cost from node i to j for each vehicle of carrier k

\( \tau_{ij}^{k} \) :

the traveling time from node i to node j for each vehicle of the carrier k

\( q_{ij}^{k} \) :

quantity of freight transported through arc (i, j) by vehicles of carrier k

\( x_{ij}^{k} \) :

the number of times that arc (i, j) is visited by vehicles of carrier k

y lk :

1 if request l is reallocated to carrier k, 0 otherwise

\( t_{i}^{k} \) :

the time at which a vehicle of carrier k leaves node i

With the notations, the mixed integer programming model is given as (1629).

Model CRRP:

$$ Z^{II} = {\text{Max}}\left( {\sum\limits_{k = 1}^{K} {\sum\limits_{l = 1}^{L} {p_{l} \cdot y_{lk} \cdot (1 - \alpha_{k} )} } - \sum\limits_{k = 1}^{K} {\sum\limits_{i = 1}^{N} {\sum\limits_{j = 1,j \ne i}^{N} {c_{ij}^{k} \cdot x_{ij}^{k} } } } } \right) $$
(16)

Subject to:

$$ \sum\limits_{j = 1,j \ne i}^{N} {x_{ij}^{k} } = \sum\limits_{j = 1,j \ne i}^{N} {x_{ji}^{k} } ,\quad i = 1, \ldots ,N,\;k = 1, \ldots ,K, $$
(17)
$$ q_{ij}^{k} \le C_{k} \cdot x_{ij}^{k} ,\quad i,j = 1, \ldots ,N,\;i \ne j,\;k = 1, \ldots ,K, $$
(18)
$$ \sum\limits_{j = 1,j \ne i}^{N} {q_{ij}^{k} } - \sum\limits_{j = 1,j \ne i}^{N} {q_{ji}^{k} } = \sum\limits_{{l \in P_{i} }} {d_{l} \cdot y_{lk} } - \sum\limits_{{l \in D_{i} }} {d_{l} \cdot y_{lk} } ,\quad i = 1, \ldots ,N,\;i \ne \{ o_{k} \} ,\;k = 1, \ldots ,K, $$
(19)
$$ \sum\limits_{{j = 1,j \ne o_{k} }}^{N} {q_{{o_{k} j}}^{k} } = \sum\limits_{{l \in P_{{o_{k} }} }} {d_{l} \cdot y_{lk} } - \sum\limits_{{l \in D_{{o_{k} }} }} {d_{l} \cdot y_{lk} } ,\quad k = 1, \ldots ,K, $$
(20)
$$ \sum\limits_{{j = 1,j \ne o_{k} }}^{N} {q_{{jo_{k} }}^{k} } = 0,\quad k = 1, \ldots ,K, $$
(21)
$$ \sum\limits_{{j = 1,j \ne o_{k} }}^{N} {x_{{o_{k} j}}^{k} } \le W_{k} ,\quad k = 1, \ldots ,K, $$
(22)
$$ \sum\limits_{k = 1}^{K} {y_{lk} } \le 1,\quad l = 1, \ldots ,L, $$
(23)
$$ \sum\limits_{j = 1,j \ne i}^{N} {x_{ij}^{k} } \le 1,\quad i = 1, \ldots ,N,\;i \ne \{ o_{k} \} ,\;k = 1, \ldots ,K, $$
(24)
$$ t_{j}^{k} \ge t_{i}^{k} + \tau_{ij}^{k} \cdot x_{ij}^{k} - T_{ij} \cdot (1 - x_{ij}^{k} ),\quad i,j = 1, \ldots ,N,\;j \ne o_{k} ,\;i \ne j,\;k = 1, \ldots ,K, $$
(25)
$$ x_{ij}^{k} \ge 0,x_{ij}^{k} \in Z,\quad i,j = 1, \ldots ,N,\quad i \ne j,\;k = 1, \ldots ,K, $$
(26)
$$ y_{lk} \in \{ 0,1\} ,\quad l = 1, \ldots ,L,\;k = 1, \ldots ,K, $$
(27)
$$ q_{ij}^{k} \ge 0,q_{ij}^{k} \in R,\quad k = 1, \ldots ,K,\;i,j = 1, \ldots ,N,\;i \ne j, $$
(28)
$$ 0 \le a_{i} \le t_{i}^{k} \le b_{i} ,\quad i = 1, \ldots ,N,\;i \ne o_{k} ,\;k = 1, \ldots ,K, $$
(29)

The objective function (16) represents the surplus profit for the alliance by serving all selected requests (requests with the sum of y lk  = 1, k = 1,…,K). Constraints (17) ensure that the number of vehicles of each carrier leaving from each node (including depot nodes) is equal to the number of vehicles of the carrier arriving at the node. Constraints (18) are the vehicle capacity constraints for each carrier. Constraints (19) are the freight flow conservation equations for each carrier, assuring the flow balance at each node (except for all depot nodes) for each carrier. Constraints (20) and (21) are the flow conservation equations on the depot node of each carrier, which insure that only empty vehicle is returned to the depot. Constraints (22) imply that no more than W k vehicles can be used for carrier k. Constraints (23) guarantee that each request is allocated to at most one carrier. Constraints (24) ensure that each node (except for depot nodes) can be visited at most once by vehicles of each carrier. Constraints (25) indicate the time window constraints for pickup/delivery operations at all nodes for each carrier. The constraints also assure that every vehicle of each carrier starts with and ends at its depot. Constraints (29) are the time window constraints for pickup/delivery operations at all nodes for each carrier. Similar to the models presented in Sects. 4 and 5, the total profit of the alliance is the sum of the surplus profit obtained by solving model CRRP and the expected profit \( \sum\nolimits_{k = 1}^{K} {\sum\nolimits_{l = 1}^{L} {p_{l} \cdot y_{lk} \cdot \alpha_{k} } } . \)

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Dai, B., Chen, H. A multi-agent and auction-based framework and approach for carrier collaboration. Logist. Res. 3, 101–120 (2011). https://doi.org/10.1007/s12159-011-0046-9

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