New formulations for the robust vehicle routing problem with time windows under demand and travel time uncertainty

Abstract

We present new formulations for the robust vehicle routing problem with time windows (RVRPTW) under cardinality- and knapsack-constrained demand and travel time uncertainty. They are the first compact models to address the RVRPTW under travel time uncertainty while considering the knapsack uncertainty set. Moreover, our models employ different types of constraints to control time propagation based on Miller–Tucker–Zemlin and single commodity flow constraints, which are derived from the linearization of recursive equations. We develop branch-and-cut methods based on the proposed formulations, leveraging a dynamic programming algorithm to verify the robust feasibility of solutions concerning both demand and travel time uncertainty, in addition to specific and standard separation procedures from the literature. We present detailed computational results on RVRPTW benchmark instances to compare the performance of our models and algorithms. Furthermore, we evaluate the impact and advantages of implementing each studied uncertainty set.

Appendices

Appendix 1: MTZ-based formulation for the RVRPTW under a multiple knapsack uncertainty set

To adapt the previous MTZ-based model to the multiple knapsack uncertainty set, we need new load and time variables and budget parameters. Each knapsack has a budget $\Delta ^d_l$ (demand) or $\Delta ^t_l$ (time). We provide the model for two knapsacks, and the extension to more knapsacks is trivial. The variables are now $u_{i\delta _{1}\delta _{2}}$ to represent accumulated load of the vehicle up to node i, with a total deviation $\delta _{1}$ over the demand’s nominal value for the first knapsack and $\delta _{2}$ for the second one; likewise, time variables are now $w_{i\delta _{1}\delta _{2}}$ representing the earliest possible time to start the service at node i, considering a total deviation of $\delta _{1}$ over the travel time’s nominal value for the first knapsack and $\delta _{2}$ for the second one. The model is then:

$$\begin{aligned}&\text{ min } \displaystyle \sum _{(i,j) \in A} c_{ij} x_{ij}, \end{aligned}$$

(45)

$$\begin{aligned}&\text{ s.t. } \text{(6), } \text{(7), } \text{(12), } \text{ and } \text{ to } \nonumber \\&\displaystyle u_{j \delta _{1}\delta _{2}} \ge u_{i\delta _{1}\delta _{2}}+\bar{d}_j+Q(x_{ij}-1), \ (i,j) \in A, 0\le \delta _{1}\le \Delta _1^d,0\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(46)

$$\begin{aligned}&\displaystyle u_{j \delta _{1}\delta _{2}} \ge u_{i(\delta _{1}-\hat{d}_j)\delta _{2}}+\bar{d}_j +\hat{d}_j+Q(x_{ij}-1), \ (i,j) \in A, j\in {S}^{d}_{1},j \not \in {S}^{d}_{2}, \hat{d}_j\le \delta _{1} \le \Delta _1^d, 0\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(47)

$$\begin{aligned}&\displaystyle u_{j \delta _{1}\delta _{2}} \ge u_{i\delta _{1}(\delta _{2}-\hat{d}_j)}+\bar{d}_j +\hat{d}_j+Q(x_{ij}-1), \ (i,j) \in A, j\not \in {S}^{d}_{1}, j \in {S}^{d}_{2}, 0\le \delta _{1} \le \Delta _1^d, \hat{d}_j\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(48)

$$\begin{aligned}&\displaystyle u_{j \delta _{1}\delta _{2}} \ge u_{i(\delta _{1}-\hat{d}_j)(\delta _{2}-\hat{d}_j)}+\bar{d}_j +\hat{d}_j+Q(x_{ij}-1), \ (i,j) \in A, j\in {S}^{d}_{1},j \in {S}^{d}_{2}, \hat{d}_j\le \delta _{1} \le \Delta _1^d, \hat{d}_j\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(49)

$$\begin{aligned}&\displaystyle u_{j \Delta _1^d\delta _{2}} \ge u_{i(\Delta _1^d-\lambda )\delta _{2}}+\bar{d}_j +\lambda +Q(x_{ij}-1), \ (i,j) \in A, j\in {S}^{d}_{1}, j \not \in {S}^{d}_{2}, 0\le \lambda \le \hat{d}_j, 0\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(50)

$$\begin{aligned}&\displaystyle u_{j \delta _{1}\Delta _2^d} \ge u_{i\delta _{1}(\Delta _2^d-\lambda )}+\bar{d}_j +\lambda +Q(x_{ij}-1), \ (i,j) \in A, j\not \in {S}^{d}_{1}, j \in {S}^{d}_{2}, 0\le \lambda \le \hat{d}_j, 0\le \delta _{1}\le \Delta _1^d, \end{aligned}$$

(51)

$$\begin{aligned}&\displaystyle u_{j \delta _{1}\Delta _2^d} \ge u_{i(\delta _{1}-\lambda )(\Delta _2^d-\lambda )}+\bar{d}_j +\lambda +Q(x_{ij}-1), \ (i,j) \in A, j\in {S}^{d}_{1}, j \in {S}^{d}_{2}, \lambda \le \hat{d}_j, \end{aligned}$$

(52)

$$\begin{aligned}&\displaystyle u_{j\Delta _1^d\Delta _2^d} \le Q, \ j\in N, \end{aligned}$$

(53)

$$\begin{aligned}&\displaystyle w_{j \delta _{1}\delta _{2}} \ge w_{i(\delta _{1}-\lambda )\delta _{2}}+\bar{t}_{ij}+s_i +\lambda +b_{n+1}(x_{ij}-1), \ (i,j) \in A, (i,j)\in {S}^{t}_{1}, (i,j) \not \in {S}^{t}_{2}, 0\le \lambda \le \min \{\hat{t}_{ij}, \delta _{1}\}\le \Delta _1^t, 0\le \delta _{2}\le \Delta _2^t, \end{aligned}$$

(54)

$$\begin{aligned}&\displaystyle w_{j \delta _{1}\delta _{2}} \ge w_{i\delta _{1}(\delta _{2}-\lambda )}+\bar{t}_{ij}+s_i +\lambda +b_{n+1}(x_{ij}-1), \ (i,j) \in A, (i,j)\not \in {S}^{t}_{1}, (i,j) \in {S}^{t}_{2}, 0\le \delta _{1}\le \Delta _1^t, \lambda \le \min \{\hat{t}_{ij},\delta _{2}\}\le \Delta _2^t, \end{aligned}$$

(55)

$$\begin{aligned}&\displaystyle w_{j \delta _{1}\delta _{2}} \ge w_{i(\delta _{1}-\lambda )(\delta _{2}-\lambda )}+\bar{t}_{ij}+s_i +\lambda +b_{n+1}(x_{ij}-1), \ (i,j) \in A, (i,j)\in {S}^{t}_{1}, (i,j) \in {S}^{t}_{2}, 0\le \lambda \le \hat{t}_{ij},\lambda \le \delta _{1}\le \Delta _1^t, \lambda \le \delta _{2}\le \Delta _2^t, \end{aligned}$$

(56)

$$\begin{aligned}&\displaystyle a_j\le w_{j \delta _{1}\delta _{2}}\le b_j, \ (i,j) \in A, 0\le \delta _{1} \le \Delta ^t_1, 0\le \delta _{2} \le \Delta ^t_2, \end{aligned}$$

(57)

$$\begin{aligned}&u_{i\delta _{1}\delta _{2}} \ge 0, \ i\in N, 0\le \delta _{1} \le \Delta ^d_1, 0\le \delta _{2} \le \Delta ^d_2, \end{aligned}$$

(58)

$$\begin{aligned}&w_{i\delta _{1}\delta _{2}} \ge 0, \ i\in N, 0 \le \delta _{1} \le \Delta ^t_1, 0 \le \delta _{2} \le \Delta ^t_2. \end{aligned}$$

(59)

Similarly to the previous models, the objective function (45) seeks to minimize the total traveling costs. Constraints (46)–(52) compute the demand for the worst case and forbid subtours. The vehicle capacity is ensured by constraints (53). Constraints (54)–(56) are similar to (50)–(52), but for the travel time. The time windows constraints are imposed by (57) and the domains of the variables are defined in (58)–(59). Note that we can easily extend this model for k knapsacks by adding indices from $\delta _1$ to $\delta _k$.

Appendix 2: Commodity-flow formulation for the RVRPTW under a multiple knapsack uncertainty set

Similarly to how it was done for the MTZ-based model, we extend formulation (32)–(38) to consider multiple knapsacks. Again, we present a formulation containing two knapsacks as the extension to k knapsacks is trivial, but the number of constraints grows quickly. This formulation uses the same sets and parameters as the MTZ-based model, while the load and time variables are now $f_{ij\delta _{1}\delta _{2}}$ and $g_{ij\delta _{1}\delta _{2}}$ with similar interpretations. The resulting model is given by:

$$\begin{aligned}&\text{ min } \displaystyle \sum _{(i,j) \in A} c_{ij} x_{ij}, \end{aligned}$$

(60)

$$\begin{aligned}&\text{ s.t. } \text{(6), } \text{(7), } \text{(12), } \text{ and } \text{ to } \nonumber \\&\displaystyle \sum _{j:(i,j) \in A}f_{ij\delta _{1}\delta _{2}} \ge \bar{d}_i+\sum _{h:(h,i) \in A}f_{hi\delta _{1}\delta _{2}}, \ i \in N, 0\le \delta _{1}\le \Delta _1^d, 0\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(61)

$$\begin{aligned}&\displaystyle \sum _{j:(i,j) \in A}f_{ij\delta _{1}\delta _{2}} \ge \bar{d}_i+\hat{d}_i +\sum _{h:(h,i) \in A}f_{hi(\delta _{1}-\hat{d}_i)\delta _{2}}, \ i \in N, i\in S^{d}_{1},i \not \in S^{d}_{2}, \hat{d}_i\le \delta _{1}\le \Delta _1^d, 0\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(62)

$$\begin{aligned}&\displaystyle \sum _{j:(i,j) \in A}f_{ij\delta _{1}\delta _{2}} \ge \bar{d}_i+\hat{d}_i+\sum _{h:(h,i) \in A}f_{hi\delta _{1}(\delta _{2}-\hat{d}_i)}, \ i \in N, i\not \in S^{d}_{1},i \in S^{d}_{2}, 0\le \delta _{1}\le \Delta _1^d, \hat{d}_i\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(63)

$$\begin{aligned}&\displaystyle \sum _{j:(i,j) \in A}f_{ij\delta _{1}\delta _{2}} \ge \bar{d}_i+\hat{d}_i+\sum _{h:(h,i) \in A}f_{hi(\delta _{1}-\hat{d}_i)(\delta _{2}-\hat{d}_i)}, \ i \in N, i\in S^{d}_{1},i \in S^{d}_{2}, 0\le \delta _{1}\le \Delta _1^d, \hat{d}_i\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(64)

$$\begin{aligned}&\displaystyle \sum _{j:(i,j) \in A}f_{ij\Delta _1^d\delta _{2}} \ge \bar{d}_i+\lambda +\sum _{h:(h,i) \in A}f_{hi(\Delta _1^d-\lambda )\delta _{2}}, \ i \in N, i\in S^{d}_{1},i \not \in S^{d}_{2}, 0\le \delta _{2}\le \Delta _2^d, 0\le \lambda <\hat{d}_i, \end{aligned}$$

(65)

$$\begin{aligned}&\displaystyle \sum _{j:(i,j) \in A}f_{ij\delta 1\Delta _2^d} \ge \bar{d}_i+\lambda +\sum _{h:(h,i) \in A}f_{hi\delta _{1}(\Delta _2^d-\lambda )}, \ i \in N, i\not \in S^{d}_{1},i \in S^{d}_{2}, 0\le \delta _{1}\le \Delta _1^d, 0\le \lambda <\hat{d}_i, \end{aligned}$$

(66)

$$\begin{aligned}&\displaystyle \sum _{j:(i,j) \in A}f_{ij\Delta _1^d\Delta _2^d} \ge \bar{d}_i+\lambda +\sum _{h:(h,i) \in A}f_{hi(\Delta _1^d-\lambda )(\Delta _2^d-\lambda )}, \ i \in N, i\in S^{d}_{1},i \in S^{d}_{2},0\le \lambda <\hat{d}_i, \end{aligned}$$

(67)

$$\begin{aligned}&\displaystyle d_ix_{ij} \le f_{ij\delta _1\delta _2} \le (Q-d_j) x_{ij}, \ (i,j) \in A, 0\le \delta _{1}\le \Delta _1^d,0\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(68)

$$\begin{aligned}&\displaystyle \sum _{j:(i,j) \in A}g_{ij\delta _{1}\delta _{2}} \ge \sum _{\begin{array}{c} h:(h,i) \in A\\ \lambda \le \hat{t}_{hi} \end{array}} (g_{hi(\delta _{1}-\lambda )\delta _{2}}+(\bar{t}_{hi}+\lambda )x_{hi})+s_i, \ i \in N,(i,j)\in S^{t}_{1},(i,j) \not \in S^{t}_{2}, 0\le \lambda \le \delta _{1} \le \Delta _1^t, 0\le \delta _{2} \le \Delta _2^t, \end{aligned}$$

(69)

$$\begin{aligned}&\displaystyle \sum _{j:(i,j) \in A}g_{ij\delta _{1}\delta _{2}} \ge \sum _{\begin{array}{c} h:(h,i) \in A\\ \lambda \le \hat{t}_{hi} \end{array}} (g_{hi\delta _{1}(\delta _{2}-\lambda )}+(\bar{t}_{hi}+\lambda )x_{hi})+s_i, \ i \in N,(i,j)\not \in S^{t}_{1},(i,j) \in S^{t}_{2}, 0\le \delta _{1} \le \Delta _1^t, 0\le \lambda \le \delta _{2} \le \Delta _2^t, \end{aligned}$$

(70)

$$\begin{aligned}&\displaystyle \sum _{j:(i,j) \in A}g_{ij\delta _{1}\delta _{2}} \ge \sum _{\begin{array}{c} h:(h,i) \in A\\ \lambda \le \hat{t}_{hi} \end{array}} (g_{hi(\delta _{1}-\lambda )(\delta _{2}-\lambda )}+(\bar{t}_{hi}+\lambda )x_{hi})+s_i, \ i \in N,(i,j)\in S^{t}_{1},(i,j) \in S^{t}_{2}, 0\le \lambda \le \delta _{1} \le \Delta _1^t, 0\le \lambda \le \delta _{2} \le \Delta _2^t, \end{aligned}$$

(71)

$$\begin{aligned}&\displaystyle (a_j+s_j)x_{ij}\le g_{ij\delta _{1}\delta _{2}}\le (b_j+s_j)x_{ij}, \ (i,j) \in A, 0\le \delta _{1} \le \Delta _1^t, 0\le \delta _{2} \le \Delta _2^t \end{aligned}$$

(72)

$$\begin{aligned}&x_{ij} \in \{0, 1\}, \ (i,j) \in A, \end{aligned}$$

(73)

$$\begin{aligned}&f_{ij\delta _{1}\delta _{2}} \ge 0, \ (i,j) \in A, 0\le \delta _{1}\le \Delta _1^d, 0\le \delta _{2}\le \Delta _2^d, \end{aligned}$$

(74)

$$\begin{aligned}&g_{ij\delta _{1}\delta _{2}} \ge 0, \ (i,j) \in A, 0\le \delta _{1}\le \Delta _1^t, 0\le \delta _{2}\le \Delta _2^t. \end{aligned}$$

(75)

The interpretation of this model is similar to the previous one, but the variables are related to the arcs and not the nodes.

Appendix 3: Results for customer-per-route ratio

In this section, we present the average results for the customer-per-route ratio discussed in Sect. 5.4. The results are summarized in Table 10, which shows the average ratio for each budget and deviation (Dev) configuration for each uncertainty set. The budgets are displayed in the format $[\Gamma ^d,\Gamma ^t]$ / $[\Delta ^d,\Delta ^t]$ according to the uncertainty set.

Table 10 Average customer-per-route ratio for each budget and deviation level