Detour-to-Depot Recourse Policy in VRPSD

• Detour-to-depot recourse policy is a VRPSD rule where vehicles return to the depot only after failing to meet customer demand, avoiding preventive restocking.
• The approach enables route decomposition by calculating recourse costs on each fixed route, which aids disaggregated integer L-shaped optimization methods.
• Recent studies show that ensuring superadditivity under concatenation and effective cut strategies yields significant computational improvements.

Searching arXiv for the cited VRPSD papers to ground the article and verify metadata. The detour-to-depot recourse policy is a recourse rule for the vehicle routing problem with stochastic demands (VRPSD) in an a priori route-design framework in which routes are planned first and customer demands are revealed only upon arrival. Under this policy, if the vehicle cannot satisfy a customer demand with its residual load, it must return to the depot, reload, and then go back to the customer; unlike the optimal restocking policy, there is no preventive depot return between customers, and depot visits happen only after a stockout or failure (Legault et al., 7 Aug 2025). Within the recent VRPSD literature, the policy is important both as an operational model of reactive restocking and as the setting in which the disaggregated integer L-shaped method was originally developed and later rigorously re-examined (Parada et al., 2022).

1. Definition within the VRPSD framework

In the VRPSD under the detour-to-depot policy, the first-stage solution fixes routes, while the second stage does not redesign routes but handles uncertainty by returning to the depot to restock whenever a vehicle cannot satisfy the next realized customer demand with its remaining capacity (Parada et al., 2022). The operational restriction is explicit: restocking trips occur only on failure. The formulation also excludes a depot return followed by an immediate move to the next customer in the case of exact stock-out (Parada et al., 2022).

For a route $(0,p,0)$ with customer sequence $p=(i_1,\dots,i_t)$, vehicle capacity $Q$, random demand $\xi_i$ at customer $i$, and failure cost $c^F_{i_j}$ at customer $i_j$, the expected recourse cost under DTD is defined by

$\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$

The probability term

$$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$

is the probability that, by the time the vehicle reaches customer $i_j$, cumulative demand has crossed the $p=(i_1,\dots,i_t)$0-th multiple of capacity, so the vehicle has already had to restock $p=(i_1,\dots,i_t)$1 times and then fails again at $p=(i_1,\dots,i_t)$2 (Legault et al., 7 Aug 2025).

The 2022 formulation expresses the same mechanism for orientation $p=(i_1,\dots,i_t)$3 as

$p=(i_1,\dots,i_t)$4

Here the $p=(i_1,\dots,i_t)$5-th restocking trip occurs at customer $p=(i_1,\dots,i_t)$6 precisely when cumulative demand first crosses the threshold $p=(i_1,\dots,i_t)$7 between customers $p=(i_1,\dots,i_t)$8 and $p=(i_1,\dots,i_t)$9 (Parada et al., 2022). If all demands satisfy $Q$0 almost surely, the inner sum can be truncated at $Q$1; otherwise it is effectively infinite but in practice truncated once probabilities become negligible (Parada et al., 2022).

A further modeling point is that a route may be evaluated in both orientations because the first-stage model does not encode direction, so

$Q$2

This route-wise structure is central to subsequent decomposition methods (Parada et al., 2022).

2. Route decomposition and the recourse structure

The DTD policy induces a recourse cost that decomposes by route. For a first-stage solution $Q$3 with route set $Q$4,

$Q$5

so the expected second-stage cost is evaluated route by route rather than through route redesign (Parada et al., 2022). This decomposition is what makes the VRPSD particularly well suited to integer L-shaped methods and, more specifically, to recourse disaggregation by route or by customer (Parada et al., 2022).

In the DTD setting, the failure cost at a customer is a round trip to the depot and back. In the 2022 paper this is written as $Q$6 when the failure occurs at customer $Q$7 (Parada et al., 2022). In the 2025 treatment, the same event is represented generically by $Q$8, emphasizing that the recourse term is the expected number of failures at customer $Q$9 multiplied by the corresponding failure cost (Legault et al., 7 Aug 2025).

This route-wise recourse function is the object on which structural properties such as monotonicity over subsequences and superadditivity under concatenation are studied. The later literature shows that these properties are not interchangeable: one is a local comparison against subsequences, whereas the other is a decomposition property under path concatenation (Legault et al., 7 Aug 2025).

3. Monotonicity and its limits under DTD

A central issue in the original disaggregated integer L-shaped analysis was monotonicity. In the 2022 paper, the method requires monotonicity in the sense that removing customers from a route should not increase expected recourse (Parada et al., 2022). The corresponding result is

$\xi_i$0

for any subpath $\xi_i$1 of a path $\xi_i$2, provided the customers on $\xi_i$3 satisfy the monotonicity condition (Parada et al., 2022).

That monotonicity condition is stated on customer sets. A set $\xi_i$4 respects the condition if for any distinct $\xi_i$5 and any $\xi_i$6,

$\xi_i$7

for all $\xi_i$8, where $\xi_i$9. Equivalently,

$i$0

The instance property is that every customer set with total expected demand no more than $i$1 satisfies this condition (Parada et al., 2022).

The same paper also establishes that DTD recourse is not monotone in general. It provides counterexamples showing that the claim can fail both when the total expected demand exceeds capacity and even when it does not (Parada et al., 2022). Accordingly, monotonicity is not an unconditional property of DTD; it holds only for certain demand families and parameter regimes (Parada et al., 2022).

The sufficient distributional conditions stated are precise:

• Poisson demands: if $i$2 are independent and $i$3, then the monotonicity condition holds (Parada et al., 2022).
• Normal demands: if $i$4 are independent with integer means, a common coefficient of dispersion $i$5, and $i$6, then monotonicity holds; the support is approximated as nonnegative in practice (Parada et al., 2022).
• Binomial demands: if $i$7 share a common success probability $i$8, and $i$9, monotonicity holds (Parada et al., 2022).
• Erlang demands: if $c^F_{i_j}$0 share a common rate $c^F_{i_j}$1, and $c^F_{i_j}$2, monotonicity holds (Parada et al., 2022).
• Negative binomial demands: if $c^F_{i_j}$3 share a common success probability $c^F_{i_j}$4, and $c^F_{i_j}$5, monotonicity holds (Parada et al., 2022).

The paper also gives a more general sufficient condition via an i.i.d.-sum representation. Defining

$c^F_{i_j}$6

a sufficient condition for monotonicity is

$c^F_{i_j}$7

This structural statement underpins the family-specific results (Parada et al., 2022).

4. DL-shaped reformulation and the correction of the original validity argument

The disaggregated integer L-shaped method replaces a single recourse variable $c^F_{i_j}$8 with customer-specific variables,

$c^F_{i_j}$9

so that cuts can act on more structured subsets of solutions than traditional aggregate cuts (Legault et al., 7 Aug 2025). In the 2025 paper, the DL-shaped master problem is written as

$i_j$0

subject to the routing constraints and two cut families, P-cuts and S-cuts (Legault et al., 7 Aug 2025).

For any feasible path $i_j$1, the path cut is

$i_j$2

If the route contains path $i_j$3 as a subpath, the factor becomes $i_j$4; otherwise the right-hand side is nonpositive and the cut is inactive (Legault et al., 7 Aug 2025). For a customer subset $i_j$5, the set cut is

$i_j$6

These are lower-bounding functionals that impose a lower bound on recourse when the solution contains enough edges inside $i_j$7 to form the required number of subpaths (Legault et al., 7 Aug 2025).

The original DL-shaped paper had argued that monotonicity over subsequences was sufficient for validity. The later paper shows that this is not sufficient (Legault et al., 7 Aug 2025). The flaw is structural: a path may satisfy

$i_j$8

yet fail

$i_j$9

Thus subsequence monotonicity does not guarantee the stronger decomposition property needed for a valid customer-wise reformulation (Legault et al., 7 Aug 2025).

The corrected theorem is exact: $\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$0 Equivalently, the necessary and sufficient condition is

$\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$1

This establishes the true validity criterion for DTD-based disaggregation (Legault et al., 7 Aug 2025).

The proof is two-sided. If superadditivity fails, then some P-cut combination can force

$\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$2

for a feasible integer solution, so the master problem overestimates the true objective and is invalid (Legault et al., 7 Aug 2025). If superadditivity holds, then for any integer solution composed of routes $\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$3,

$\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$4

and customer-wise values can be assigned through incremental contributions

$\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$5

These increments are nonnegative under superadditivity, and telescoping gives

$\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$6

Hence the formulation is exact (Legault et al., 7 Aug 2025).

5. Superadditivity under concatenation and the status of DTD

The key structural condition introduced in the 2025 paper is superadditivity under concatenation: $\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$7 Its interpretation is that the recourse cost of the combined path is at least the sum of the recourse costs of the two parts (Legault et al., 7 Aug 2025). In the DL-shaped context, this is exactly the property needed to ensure that customer-wise decomposition does not overestimate the true recourse when one path is viewed as a concatenation of smaller pieces (Legault et al., 7 Aug 2025).

For DTD, the paper proves the following conditional statement: $\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$8 This is the rigorous correction of the earlier result. The earlier argument appealed directly to monotonicity over subsequences; the corrected argument shows that the relevant conclusion is superadditivity, and then proves DTD superadditivity from the monotonicity property under the structural assumptions used in the original paper (Legault et al., 7 Aug 2025).

The contrast with the optimal restocking policy is explicit. The same 2025 paper proves

$\bar{\mathcal{Q}^{\text{DTD}_{p} := \sum_{j=1}^t \sum_{l=1}^{+\infty} \mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right] c^F_{i_j}.$9

Accordingly, the DL-shaped method is always valid for OR, with no extra assumptions, whereas for DTD validity is tied to the superadditivity condition and therefore, in the corrected proof, to the monotonicity assumptions under which superadditivity is established (Legault et al., 7 Aug 2025).

A common misconception is therefore that DTD validity rests simply on monotonicity over subsequences. The corrected result shows that this is not the actual criterion. Monotonicity may imply superadditivity in the DTD setting under the relevant assumptions, but the validity theorem itself is stated in terms of superadditivity under concatenation (Legault et al., 7 Aug 2025).

6. Lower bounds, generalized cuts, and computational role

The original 2022 DL-shaped framework introduced lower bounding functionals alongside disaggregated cuts. For a set $$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$0, the S-cut was written as

$$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$1

with $$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$2, valid if

$$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$3

for any partition of $$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$4 into feasible paths $$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$5 (Parada et al., 2022). Three lower bounds were developed: $$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$6, $$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$7, and $$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$8. The first is a single-route lower bound obtained by sorting customers by non-increasing distance to the depot and taking the resulting route $$\mathbb{P}\!\left[\sum_{k=1}^{j-1}\xi_{i_k} \le lQ < \sum_{k=1}^{j}\xi_{i_k}\right]$$9, with

$i_j$0

The second is a dynamic-programming bound, and the third is a set-covering or column-generation bound (Parada et al., 2022).

The 2025 paper extends this line by introducing edge-set cuts, or E-cuts, which generalize both P-cuts and S-cuts (Legault et al., 7 Aug 2025). For a customer set $i_j$1 and an edge subset $i_j$2, let $i_j$3 denote partitions of $i_j$4 into $i_j$5 feasible paths using only edges in $i_j$6, $i_j$7 the minimum number of vehicles needed under this restriction, and $i_j$8 the minimum recourse over such partitions. The E-cut is then

$i_j$9

The significance of E-cuts is twofold. First, the paper shows that S-cuts can be very weak when preventive depot returns are cheap, because long routes can make recourse cost vanish or nearly vanish (Legault et al., 7 Aug 2025). Second, by restricting to a carefully chosen edge subset $p=(i_1,\dots,i_t)$00, E-cuts can produce a stronger lower bound while remaining valid (Legault et al., 7 Aug 2025). The relationship to earlier cut classes is exact: S-cuts are recovered by taking $p=(i_1,\dots,i_t)$01, and P-cuts are a special case of the strongest E-cut when $p=(i_1,\dots,i_t)$02 and $p=(i_1,\dots,i_t)$03 contains exactly the edges of the path $p=(i_1,\dots,i_t)$04 (Legault et al., 7 Aug 2025).

Computationally, the 2025 study reports that, on benchmark instances adapted from Jabali et al., adding E-cuts reduced the number of unsolved instances from 53 to 6, reduced average solve time by more than a factor of 13 on solved instances, and reduced explored branch-and-bound nodes by more than a factor of 7 (Legault et al., 7 Aug 2025). The paper also states that the new DL-shaped algorithm outperforms the state-of-the-art integer L-shaped methods, is especially effective on instances with few long routes, and solves several open instances to optimality, including 14 single-vehicle instances, which constitute the most challenging variant of the problem (Legault et al., 7 Aug 2025).

These results suggest a broader methodological point. In the DTD setting, the recourse policy creates a route-wise structure, but the practical effectiveness of exact methods depends not only on the structural validity of disaggregation but also on the strength of lower bounds and cut families used in the master problem (Parada et al., 2022, Legault et al., 7 Aug 2025).