Winner Determination with Sustainable-Flexible Considerations Under Demand Uncertainty in Transportation Service Procurement Auctions

Abstract

Sustainability and flexibility are two main factors not being investigated explicitly by existing winner determination literature. From a fourth party logistics (4PL) provider’s point of view, an innovative sustainable-flexible winner determination problem under uncertain demand is particularly studied in transportation services procurement auctions. Based on a multi-attribute decision-making method, a linear constraint related to each bidder’s sustainability score and flexibility score can be constructed, and then we integrate an outside option policy to formulate a two-stage stochastic sustainable-flexible winner determination model. Subsequently, we develop an approximation approach to solve the model based on the principles of dual decomposition Lagrangian relaxation and the sample average approximation. Using an established generator to obtain random instances, the effectiveness and applicability of this research could be verified by conducting numerical experiments. Also, managerial insights can be obtained to provide decision support for running an efficient sustainable-flexible logistics system by using a Chinese 4PL firm’s real data.

Appendix Data of Different Problems

Small-Scale Problem

Set $\mid I \mid = 5$, $\mid J \mid = 10$, $\mid K_j \mid = 2$, $N_{\max }=10$, $N_{\min }=0$, $\alpha =0.6$, $\alpha '=0.8$, $\beta =0.7$, $\beta '=0.8$, $T=10000000$, $\mu _{\varvec{\xi }} = 250$, $\sigma _{\varvec{\xi }} = 82.603$, $M=20$, $N=20$, and $\hat{N}=1000$. Other parameters follow uniform distributions, which could be characterized by ${\text {U}}[a,b]$. Here, we have $b_{jk} \sim {\text {U}}[100,250]$, $v_{j} \sim {\text {U}}[2000,3000]$, $v'_{j}=0.8v_{j}$, $US_{jk} \sim {\text {U}}[200,300]$, $LS_{jk} \sim {\text {U}}[50,100]$, $\vartheta _j \sim {\text {U}}[0.5,1]$, $\nu _j \sim {\text {U}}[0.5,1]$, $i \in I,~j \in J$.

Each 3PLs’s bidding package can be expressed by [no. of 3PL, {package 1}, {package 2}], that is, [1, {1}, {1, 2}], [2, {2}, {2, 3}], [3, {3}, {3, 4}], [4, {4}, {4, 5}], [5, {5}, {1, 5}], [6, {1, 3}, {1, 2}], [7, {2, 3}, {2, 5}], [8, {3, 4}, {3, 5}], [9, {5}, {1}], [10, {2}, {4}].

Medium-Scale Problem

Set $\mid I \mid = 20$, $\mid J \mid = 40$, $\mid K_j \mid \le 5$, $N_{\max }=40$, $N_{\min }=0$, $\alpha =0.6$, $\alpha '=0.8$, $\beta =0.7$, $\beta '=0.8$, $T=10000000$, $\mu _{\varvec{\xi }} = 600$, $\sigma _{\varvec{\xi }} = 288.675$, $M=20$, $N=20$, and $\hat{N}=1000$. Other parameters follow uniform distributions, which could be characterized by ${\text {U}}[a,b]$. Here, we have $b_{jk} \sim {\text {U}}[200,300]$, $v_{j} \sim {\text {U}}[2000,5000]$, $v'_{j}=0.7v_{j}$, $US_{jk} \sim {\text {U}}[300,400]$, $LS_{jk} \sim {\text {U}}[50,100]$, $\vartheta _j \sim {\text {U}}[0.5,1]$, $\nu _j \sim {\text {U}}[0.5,1]$, $i \in I,~j \in J$.

The potential 3PLs’ bidding packages can be given by [1, {2, 3}, {1, 11, 12}, {1, 15, 16}, {1, 17}, {1, 19}], [2, {1, 2}, {1, 2, 11, 12}, {2, 3, 15, 16}, {4, 5}, {16, 17, 18}], [3, {2, 3}, {4, 5, 6}, {19, 20}, {1, 19, 20}, {3, 17, 18}], [4, {5, 6}, {4, 5, 6}, {5, 6, 15}, {11, 12, 15}, {16, 17, 18}], [5, {3, 4, 5}, {2, 3, 11}, {4, 5, 6}, {11, 15, 16}, {18, 19, 20}], [6, {11}, {17}, {20}, {5, 6}, {3, 4}], [7, {11, 12}, {13, 14}, {7, 13}, {19}, {20}], [8, {7}, {8}, {11}, {16}, {19}], [9, {5, 6}, {3}, {4}, {15, 16}, {1, 3}], [10, {11, 12}, {15, 16}, {17, 18, 19}, {19}, {20}], [11, {16}, {17}, {18}, {2, 15}, {1, 16}], [12, {3, 4, 5}, {15, 16, 17}, {11, 12, 15}, {17, 18, 19}, {2, 4, 5}], [13, {15}, {11}, {12}, {12, 15}, {17}], [14, {3}, {4}], [15, {16}, {17}], [16, {9, 10}, {7, 8}], [17, {13, 14}], [18, {7, 9}], [19, {17}, {18}], [20, {5}, {6}], [21, {2, 3}, {3}], [22, {16, 17}], [23, {18, 19}], [24, {20}], [25, {19}], [26, {11}], [27, {12}], [28, {1, 4}, {5, 6}], [29, {13}, {14}], [30, {16, 17, 18}], [31, {18, 19}], [32, {7, 8}, {9}], [33, {1, 2}, {1, 16}, {1, 17}], [34, {2, 3}, {2, 16}], [35, {5, 8}], [36, {17, 18}, {19}], [37, {5, 6}, {11, 12, 15}], [38, {17}, {10}], [39, {15, 19}, {9, 10, 13, 14}], [40, {10}, {9, 13, 14}].

Large-Scale Problems

Set $\mid I \mid = 40$, $\mid J \mid = 80$, $\mid K_j \mid = 5$, $N_{\max }=80$, $N_{\min }=0$, $\alpha =0.6$, $\alpha '=0.8$, $\beta =0.7$, $\beta '=0.8$, $T=10000000$, $\mu _{\varvec{\xi }} = 10000$, $\sigma _{\varvec{\xi }} = 288.675$, $M=20$, $N=20$, and $\hat{N}=1000$. Other parameters follow uniform distributions, which could be characterized by ${\text {U}}[a,b]$. Here, we have $v_{j} \sim {\text {U}}[10000,30000]$, $v'_{j}=0.7v_{j}$, $US_{jk} \sim {\text {U}}[150,200]$, $LS_{jk} \sim {\text {U}}[50,100]$, $\phi _j \sim {\text {U}}[0.5,1]$, $\varphi _j \sim {\text {U}}[0.5,1]$, $i \in I,~j \in J$. The potential 3PLs’ bidding packages and the corresponding bid prices could be generated randomly by using CATS.