Algorithmic Mechanism Design for Collaboration in Large-Scale Transportation Networks

Abstract

The importance of collaborative logistics is getting widely recognized in recent years. However, strategic revealing of private information and disagreements on how savings are split would make any efforts of collaboration unsuccessful. The large-scale nature of real-life transportation networks further complicates the implementation of collaboration, as the associated optimization problems are usually NP-hard. The academic community has developed a variety of methodologies by using operations research tools and algorithmic mechanism design to resolve these issues. We summarize the state-of-the-art progress in the literature and introduce the iterative mechanism design theories. The iterative mechanism design models consider private information and propose decentralized approximation algorithms to achieve a system-wide objective while eliciting truthful information through the iterations. Hence, iterative mechanism design effectively reduces computational and communication difficulties. We then apply the methodology to a truckload pickup-and-delivery collaboration problem as an example. Numerical results on large-scale instances are reported, verifying the effectiveness of the methodologies.

Appendices

Appendix 1: The TL Pickup-and-Delivery Model

We model an individual carrier’s TL pickup-and-delivery problem using a two-index formulation.

For ease of exposition, we introduce a dummy request (D _i, D _i) denoted by 0 for each carrier i, i.e., o ₀ = D _i and d ₀ = D _i. Define binary variable $x_{kl}^{i}=1$, if a truck of carrier i travels from the delivery node d _k of request k to the pickup node o _l of request l, where k ≠ l. That is, request l is picked up by a truck which has just delivered request k or left the depot.

Define $f_{l}^{i}$ to be the travel distance from the depot D _i to the delivery node d _l of request l and $t_{l}^{i}$ to be the delivery time of request l by a truck of carrier i. For convenience, the distance from d _k to o _l is redefined as

$$\displaystyle \begin{aligned} a_{kl}= \left\{ \begin{array}{ll} a_{(d_{k},o_{l})}, & \text{if }d_{k}\neq o_{l}\text{;} \\ 0, & \text{if }d_{k}=o_{l}\text{.} \\ \end{array} \right. \end{aligned}$$

Moreover, write $a_{l}:=a_{(o_{l}, d_{l})}$. The travel time between requests k and l is then equal to $T_{kl}:= \frac {a_{kl}} {\gamma }$.

Each variable $x_{kl}^{i}$ is associated with a cost parameter $c_{kl}^{i}=c_{i}\cdot a_{kl}$, representing the traveling cost. By default, variable $x_{kl}^{i}$ and its cost parameter $c_{kl}^{i}$ are valid only for k ≠ l through the paper and thus we do not particularly state k ≠ l any more in the models. After arriving at o _l, the truck further travels a distance of $a_{(o_{l}, d_{l})}$ to deliver request l. The profit in the delivery of request l ∈ L _i excluding the empty traveling cost is $p_{l}^{i}:=r_{l}^{i}-c_{i} \cdot a_{(o_{l}, d_{l})}$.

If not participating in the collaboration, carrier i must deliver all the freight requests l ∈ L _i using his own trucks. The individual optimization problem of carrier i, denoted by $P_i^0$, is formulated as below.

$$\displaystyle \begin{aligned} \begin{array}{rcl} \varPi_{i}^{*}(L_{i}) = \max &\displaystyle &\displaystyle \sum_{l \in L_{i}} p_{l}^{i}- \rho_{i}\cdot \sum_{k\in L_{i}\cup \{0\}}\sum_{l\in L_{i}\cup \{0\}} c_{kl}^{i}x_{kl}^{i} \\ {} (P_{i}^{eval})\quad \text{s.t.} &\displaystyle &\displaystyle \sum_{k\in L_{i}\cup\{0\}}x_{kl}^{i}=1, \quad \forall\, l\in L_{i}; \end{array} \end{aligned} $$

(10)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle \sum_{k\in L_{i}\cup\{0\}}x_{kl}^{i}-\sum_{k\in L_{i}\cup\{0\}} x_{lk}^{i}=0, \quad \forall\, l\in L_{i}\cup\{0\}; \end{array} \end{aligned} $$

(11)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle f_{0}^{i}=0; \end{array} \end{aligned} $$

(12)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle f_{k}^{i}+a_{kl}+a_{l}-M \cdot (1-x_{kl}^{i}) \leq f_{l}^{i}, \forall\,k\in L_{i}\cup\{0\}, l\in L_{i};\qquad \end{array} \end{aligned} $$

(13)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle f_{l}^{i}+a_{l0}-M \cdot (1-x_{l0}^{i})\leq F, \quad \forall\, l\in L_{i}; \end{array} \end{aligned} $$

(14)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle f_{l}^{i}\leq F, \quad \forall\, l\in L_{i}; \end{array} \end{aligned} $$

(15)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle t_{0}^{i}=0; \end{array} \end{aligned} $$

(16)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle t_{k}^{i}+T_{kl}+T_{l}^{S}-M(1-x_{kl}^{i})\leq t_{l}^{i}, \; \forall\, k\in L_{i}\cup \{0\}, l\in L_{i}; \end{array} \end{aligned} $$

(17)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle &\displaystyle T_{l}^{P}+T_{l}^{S}\leq t_{l}^{i} \leq T_{l}^{D}, \quad \forall\, l\in L_{i}; \end{array} \end{aligned} $$

(18)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle x_{kl}^{i} \in \{0, 1\}, \quad \forall\, k,l\in L_{i}\cup\{0\}; \\ &\displaystyle &\displaystyle f_{l}^{i}, t_{l}^{i}\geq 0, \quad \forall\, l\in L_{i}. {} \end{array} \end{aligned} $$

(19)

In the above model, M is a sufficiently large positive number. Constraints (10) require that each request must be served. Constraints (11) impose the degree balance condition for each request and the depot, implying that if a truck enters the pickup node of a request, it must leave from delivery node of this request. Constraints (12)–(15) track the travel distance of each truck, where the big number M is used to maintain the feasibility, and the total travel distance cannot exceed F. Constraints (13) imply the subtour-elimination condition Ropke and Pisinger [31], Li et al. [24], and hence the route of each truck must start from the depot and end at the depot. Constraint (16) sets the departure time of a truck starting from the depot, (17) enforces the time precedence relationship between consecutively served requests, and (18) states the feasible time for serving request l. Constraints (19) impose the integrality of variables $x_{kl}^{i}$.

The objective is to maximize the profit of carrier i. Since each request must be fulfilled and the total revenue is a constant, the objective is equivalent to minimizing the total empty traveling distance. After delivering request k, if there is a request l with o _l = d _k, then a _kl = 0; that is, the truck does not travel empty when going to pickup request l in this case.

Appendix 2: The Optimal Request Selling Problem

Define binary variables $y_{l}^{i}$ that equals to one if request l ∈ L _i ∩ L ^b is selected to be sold and zero otherwise. Variables $x_{kl}^{i}$ and $f_{l}^{i}$ are defined as in problem ($P_{i}^{eval}$). Given the selling prices π _l and the demanded requests in L ^b at the current iteration, we formulate the seller’s problem of carrier i as below.

$$\displaystyle \begin{aligned} \begin{array}{rcl} \max &\displaystyle &\displaystyle \sum_{l\in L_{i}\setminus L^{b}} p_{l}^{i}+\sum_{l\in L_{i}\cap L^{b}} p_{l}^{i}\cdot (1-y_{l}^{i}) + \sum_{l\in L_{i}\cap L^{b}} \pi_{l} \cdot y_{l}^{i}- \\ &\displaystyle &\displaystyle \rho_{i} \cdot \sum_{k\in L_{i}\cup \{0\}} \sum_{l\in L_{i}\cup \{0\}} c_{kl}^{i} x_{kl}^{i} \\ {} (P_{i}^{sell}) \quad s.t. &\displaystyle &\displaystyle \sum_{l\in L_{i}\cap L_{j}^{*}} y_{l}^{i} \leq 1, \quad \forall\, j\in N \setminus\{i\}; \end{array} \end{aligned} $$

(20)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \sum_{k\in L_{i}\cup \{0\}} x_{kl}^{i}=1-y_{l}^{i}, \quad \forall \, l\in L_{i}\cap L^{b}; \\ &\displaystyle &\displaystyle \sum_{k\in L_{i}\cup \{0\}} x_{kl}^{i}=1, \quad \forall \, l\in L_{i}\setminus L^{b}; \\ &\displaystyle &\displaystyle ~\mbox{(11)}-\mbox{(18)}, \quad \forall\,k\in L_{i}\cup\{0\}, l\in L_{i}; \\ {} &\displaystyle &\displaystyle f_{l}^{i}\leq F\cdot (1-y_{l}^{i}), \quad \forall \, l\in L_{i}\cap L^{b}; \end{array} \end{aligned} $$

(21)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle (T_{l}^{P}{+}T_{l}^{S})\cdot (1{-}y_{l}^{i})\leq t_{l}^{i} \leq T_{l}^{D}\cdot (1{-}y_{l}^{i}), \, \forall\, l\in L_{i}\cap L^{b}; \\ &\displaystyle &\displaystyle x_{kl}^{i}, \, y_{l}^{i}\in \{0, 1\}; \\ &\displaystyle &\displaystyle f_{l}^{i}, \, t_{l}^{i}\geq 0. {} \end{array} \end{aligned} $$

(22)

In the above model, $L_{j}^{*}$ is defined as the set of requests in $L_{j}^{b}$ where carrier j is the highest bidder among all. Since each bidder can only be allocated with at most one request and the auction will allocate the request to the highest bidder, it is futile to sell more than one request to the same highest bidder. Thus, constraints (20) ensure that the seller can only sell one request to each highest bidder, and the number of selling requests is no more than the number of bidders. Constraints (21)–(22) state that if request l is sold, then the request is not fulfilled and hence $f_{l}^{i} = t_{l}^{i} =0$.