A Sampling-Based Stochastic Winner Determination Model for Truckload Service Procurement

Abstract

Many large shippers procure truckload (TL) service from carriers via a combinatorial auction. In order to determine the winners of the auction, they need to solve a combinatorial optimization problem known as winner determination problem (WDP). In practice, shippers must resolve the WDP under shipment volume uncertainty due to limited information of future demands. In this paper, we propose a sampling-based two-stage stochastic programming approach to solve WDP under shipment volume uncertainty. We propose a refined formulation of deterministic WDPs in which shortage in shipments and the associated penalty cost are explicitly modeled. We demonstrate that the refined model is more general and more feasible under uncertainty than the benchmarks. Theoretical results pertaining to problem feasibility are derived and their insights to TL service procurement are provided. We propose a sampling-based solution approach called Monte Carlo Approximation (MCA) and use numerical tests to show that MCA is numerically tractable for solving moderately sized instances of TL service procurement. Finally, we verify via Monte Carlo simulation that the solution to our proposed stochastic WDP yields lower procurement cost than the solution to the deterministic WDP.

Appendices

Appendix A: Proof of Proposition 1

Given the definition of J ⁺ and constraints (7), (13) and (18), we have $\begin{array}{ll}{\displaystyle \sum_{k\in {K}_j}{x}_{jk}}=1,\hfill & \forall j\in {J}^{+}\hfill \end{array}$. Further, ${\displaystyle \sum_{j\in {J}^{+}}{\displaystyle \sum_{k\in {K}_j}{x}_{jk}=\left|{J}^{+}\right|}}$. If N _max < |J ⁺|, constraint (14) will be violated and c-WDP becomes infeasible.

Appendix B: Proof of Proposition 2

Let’s denote the first-stage solution that is feasible to s-WDP by ${\overline{x}}_{jk},\forall j\in J,\forall k\in {K}_j$. Note that although constraints (23) and (24) do not contain a first-stage decision variable, Proposition 1 shows that a feasible first-stage solution may not exist if the parameters in these constraints are not carefully evaluated. Therefore, if there exists a feasible first-stage solution to s-WDP, we should also have ${y}_{jk}\left(\tilde{\mathbf{d}}\right)={\overline{y}}_{jk},\forall j\in J,\forall k\in {K}_j$ so that constraints (21)–(25) are satisfied. Then it is trivial that the following solution is always feasible to s-WDP:

$$ \begin{array}{ll}{x}_{jk}={\overline{x}}_{jk},\hfill & \forall j\in J,\forall k\in {K}_j\hfill \end{array} $$

$$ \begin{array}{ll}{y}_{jk}\left(\tilde{\mathbf{d}}\right)={\overline{y}}_{jk},\hfill & \forall j\in J,\forall k\in {K}_j\hfill \end{array} $$

$$ \begin{array}{ll}{\varphi}_i\left(\tilde{\mathbf{d}}\right)={\left({\tilde{d}}_i-{\displaystyle \sum_{j\in J}{\displaystyle \sum_{k\in {K}_j}{a}_{ijk}{\overline{y}}_{jk}\left(\tilde{\mathbf{d}}\right)}}\right)}^{+},\hfill & {\tilde{d}}_i\in {\varDelta}_i,\forall i\in I\hfill \end{array} $$

$$ {\theta}_i\left(\tilde{\mathbf{d}}\right)={\left({\displaystyle \sum_{j\in J}{\displaystyle \sum_{k\in {K}_j}{a}_{ijk}{\overline{y}}_{jk}\left(\tilde{\mathbf{d}}\right)-{\tilde{d}}_i}}\right)}^{+},{\tilde{d}}_i\in {\varDelta}_i,\forall i\in I $$

where

$$ {(z)}^{+}=\left\{\begin{array}{ll}z,\hfill & \mathrm{if}\kern0.5em z\ge 0\hfill \\ {}0,\hfill & \mathrm{otherwise}\hfill \end{array}\right. $$

The proposition is proven.