Computational Approaches for Grocery Home Delivery Services

We consider a fixed tour $\mathcal{A}=(0,1,\dots,n,n+1)$, where $0$ and $n+1$ are equal to the depot $d$ and $(1,\ldots,n)$ are the customers assigned to the tour. We use the concept of earliest and latest arrival time $e_{i}$ and $\ell_{i}$, as in [9], which give the earliest (latest) time at which the vehicle may arrive at customer $i$, while not violating time window and travel time constraints on the remaining tour:

	$\displaystyle e_{0}:=start_{\mathcal{A}},~$	$\displaystyle e_{j+1}:=\max\left\{B_{w(j+1)},~e_{j}+s(j)+t(j,j+1)\right\},j\in[n-1]_{0},$
		$\displaystyle e_{n+1}:=e_{n}+s(n)+t(n,n+1),$
	$\displaystyle\ell_{n+1}:=end_{\mathcal{A}},~$	$\displaystyle\ell_{j-1}:=\min\left\{E_{w(j-1)},~\ell_{j}-t(j-1,j)-s(j-1)\right\},j\in[n]\setminus\{1\},$
		$\displaystyle\ell_{0}:=\ell_{1}-t(0,1).$

Here, we assume $s(0)=s(n+1)=0$.
Following the definitions, vehicles always leave as early as possible from the depot. This generates unnecessary idle time before serving the first customer of a tour. Hence, once the delivery schedule is finalized, we alter the start times of the vehicle in order to avoid this.

A schedule $\mathcal{S}$ is feasible if all its tours are feasible. A tour $\mathcal{A}$ is feasible if it satisfies both of the following conditions:

	$\displaystyle e_{i}\leq E_{w(i)},~i\in[n]\quad\wedge\quad e_{n+1}\leq end_{\mathcal{A}},\qquad$	$\displaystyle\text{{{TFEAS($\mathcal{A}$)}}},$
	$\displaystyle\sum\limits_{i\in[n]}c(i)\leq C_{\mathcal{A}},\qquad$	$\displaystyle\text{{{CFEAS($\mathcal{A}$)}}}.$

While TFEAS($\mathcal{A}$) ensures that the arrival times at each customer assigned to tour $\mathcal{A}$ are within their assigned time windows, CFEAS($\mathcal{A}$) guarantees that the capacity of $\mathcal{A}$ is not exceeded. Note that we do not need to check for TFEAS($\mathcal{A}$) if $B_{w(i)}\leq e_{i},\ i\in[n],$ as this is ensured by the definition of $e_{i}$.

The set $\Theta(j,\mathcal{A})$ is required for the approaches applied in the Get TWs Step and defines after which customers we try to insert customer $j$ into $\mathcal{A}$ during its (pre)assigned time slot. Accordingly, we define:

		$\displaystyle\Theta(j,\mathcal{A}):=[\Theta^{-}(j,\mathcal{A}),\Theta^{+}(j,\mathcal{A})],$
where
		$\displaystyle\Theta^{-}(j,\mathcal{A}):=\min\limits_{i\in[n]_{0}}\{i\colon B_{w(j)}+s(j)\leq\ell_{i}\},$
		$\displaystyle\Theta^{+}(j,\mathcal{A}):=\max\limits_{i\in[n]_{0}}\{i\colon e_{i}+s(i)\leq E_{w(j)}\}.$

The index $\Theta^{-}(j,\mathcal{A})$ defines the first customer (or the depot) on tour $\mathcal{A}$ after which customer $j$ could potentially be inserted. Likewise, $\Theta^{+}(j,\mathcal{A})$ defines the last customer (or the depot). Clearly, if $\Theta^{-}(j,\mathcal{A})>\Theta^{+}(j,\mathcal{A})$, the insertion of $j$ during $w(j)$ is infeasible.

The feasibility of the possible insertion points $\Theta(j,\mathcal{A})$ can be checked easily with earliest and latest arrival times, e.g., see [9]. Similar to the definitions above and with the help of $e_{i}$ and $\ell_{i},\ i\in\mathcal{A}$, we define the earliest and latest arrival time $\tilde{e}_{j,i}$ and $\tilde{\ell}_{j,i}$ for inserting a new customer $j\notin\mathcal{A}$ after $i\in\Theta(j,\mathcal{A})\subseteq\mathcal{A}$ within the (pre)assigned time window $w(j)$ as follows.

	$\displaystyle\tilde{e}_{j,i}:=\max\left\{B_{w(j)},~e_{i}+s(i)+t(i,j)\right\},$
	$\displaystyle\tilde{\ell}_{j,i}:=\min\left\{E_{w(j)},~\ell_{i+1}-t(j,i+1)-s(j)\right\}.$

Thus, customer $j$ can be inserted between customers $i$ and $i+1$, such that $j$ and all subsequent customers of $\mathcal{A}$ can be served within their assigned time windows if and only if the following condition holds.

$\displaystyle\tilde{e}_{j,i}\leq\tilde{\ell}_{j,i},$

$\displaystyle\texttt{{TFEAS($j,{i},\mathcal{A}$)}},$

We refer to Figure 2 for an illustration.

Additionally, we must check if the sum of the weights of the customer orders assigned to tour $\mathcal{A}$ does not exceed the capacity $C_{\mathcal{A}}$. The insertion of $j$ into tour $\mathcal{A}$ is feasible with respect to capacity if the following condition holds:

$\displaystyle\sum\limits_{i\in[n]}c(i)+c(j)\leq C_{\mathcal{A}},$

$\displaystyle\texttt{{CFEAS($j,\mathcal{A}$)}}.$

Assuming that all earliest (latest) arrival times and the sum of order weights on a tour $\mathcal{A}$ have already been calculated, TFEAS($j,i,\mathcal{A}$) and CFEAS($j,\mathcal{A}$) allow to check the feasibility of a possible insertion into a given time window in $\mathcal{O}(1)$ time provided that the sequence of $\mathcal{A}$ (except for $j$) stays the same. If the feasibility check was successful and we decide to insert $j$, we obtain a new tour $\mathcal{\tilde{A}}=\mathcal{A}+_{i}j=(0,1,\dots,i,j,i+1,\dots,n,n+1)$. Customer $j$ is then assigned to index $i+1$, and all indices of succeeding customers are incremented by one. Clearly, the earliest and latest arrival times and the sum of order weights of the modified tour must be updated. This requires $\mathcal{O}(n)$ time [7].

In general, there exist cases where a feasible insertion of $j$ is only possible when the order of $\mathcal{A}$ is changed, which makes the problem $\mathcal{NP}$-hard.

For a given tour $\mathcal{A}$ we define the first and last customer belonging to a given time slot $u\in\mathcal{W}$ as

	$\displaystyle f(u):=\min_{i\in[n]}\left\{i:B_{u}\leq B_{w(i)}\leq E_{w(i)}\leq E_{u}\right\},$
	$\displaystyle l(u):=\max_{i\in[n]}\left\{i:B_{u}\leq B_{w(i)}\leq E_{w(i)}\leq E_{u}\right\}.$

If above sets are empty, then the indices are not defined, i.e., $[f(u),l(u)]=\emptyset$.

In case of non-overlapping time slots and if $w(j)=u,\ j\notin\mathcal{A},$ $u$ is not empty, i.e., there is at least one customer $i\in\mathcal{A}$ assigned to $u$, the following statement holds:

$$\Theta(j,\mathcal{A})\subseteq\{f(u)-1,\ldots,l(u)\}.$$

Our ANS heuristic considers two different neighborhoods for a time window $u\in\mathcal{W}$ and a tour $\mathcal{A}\in\mathcal{S}$.

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  Inside includes all operations with customers inside $u$ :
  $in(u,\mathcal{A}):=\{i\in\mathcal{A}:i\in[f(u),l(u)]\}.$

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  Outside represents operations with customers outside $u$ :
  $out(u,\mathcal{A}):=\mathcal{A}\setminus(in(u,\mathcal{A})\cup\{0,n+1\}).$

For a tour $\mathcal{A}$ and a new customer $j$ who has to be inserted into time slot $u$, we define

	$\displaystyle\chi_{u}^{-}(j,\mathcal{A})$	$\displaystyle:=\max\bigg{(}e_{f(u)},~\max_{i\in\Theta(j,\mathcal{A}),~i\leq f(u)}\tilde{e}_{j,i}\bigg{)}-B_{u},$
	$\displaystyle\chi_{u}^{+}(j,\mathcal{A})$	$\displaystyle:=E_{u}-\min\bigg{(}\ell_{l(u)},~\min_{i\in\Theta(j,\mathcal{A}),~i\geq l({u})}\tilde{\ell}_{j,i}\bigg{)}.$

The $\chi_{u}^{-}$ corresponds to the amount of time that is “lost” at the beginning of time slot ${u}$. This can be caused by the service time required for the last customer order before (outside) $u$ or the travel time needed for going from that customer to the first customer inside $u$. Similarly, $\chi_{u}^{+}$ corresponds to the loss of time at the end of time slot ${u}$ caused by the time required for traveling to the first customer after (outside) $u$ or the service time at the last customer inside $u$.

Further, we denote $\chi_{u}(j,\mathcal{A}):=\chi_{u}^{-}(\mathcal{A},j)+\chi_{u}^{+}(j,\mathcal{A})$ as the loss time of time window ${u}$. In case that $[f(u),l(u)]=\emptyset$, the loss time is given by

$$\chi_{u}(j,\mathcal{A})=\max_{i\in\Theta(j,\mathcal{A})}\big{(}(\tilde{e}_{j,i}-s_{u})+(e_{u}-\tilde{\ell}_{j,i})\big{)}.$$

Figure 4 illustrates the $\chi_{u}^{-}(j,\mathcal{A})$ and $\chi_{u}^{+}(j,\mathcal{A})$ on a tour with non-overlapping time windows. Clearly, if $\chi_{u}(j,\mathcal{A})=0$, then a violation of TFEAS($\mathcal{A}$) for $j$ can only be repaired by removing (exchanging) customers that are inside $u$.

Furthermore, we want to quantify the amount of service and travel time that is needed for inserting $j$ during time window ${u}$. For a given tour $\mathcal{A}$ the free time of time slot ${u}$ is defined as

$\displaystyle\lambda_{{u}}(\mathcal{A}):=\underbrace{\left(E_{u}-B_{u}\right)}_{(I)}-\underbrace{\sum\limits_{i=f({u})}^{l({u})-1}\big{(}s(i)+t(i,i+1)\big{)}}_{(II)},$

where $(I)$ is the length of ${u}$ and $(II)$ is the amount of service and travel time that must be handled within ${u}$. In case that the indices $f({u})$ and $l({u})$ are not defined, i.e., $in(u,\mathcal{A})=\emptyset$, term $(II)$ is set to $0$. In Figure 5, we provide an illustration of the free time.

Considering non-overlapping time windows, the insertion of customer $j$ after a customer $i\in\{{f({u})-1},\allowbreak\ldots,l({u})\}$ requires an additional amount of (travel and service) time that must be handled within time window ${u}$ and can be calculated by $\lambda_{u}(\mathcal{A})-\lambda_{{u}}(\mathcal{A}+_{i}j)$. This assumes that all customers between ${f({u})}$ and ${l({u})}$ can be moved arbitrarily (while maintaining their sequence) within time slot ${u}$, i.e., all customers assigned to $u$ can be seen as one consecutive block which length is given by $(II)$. Hence, the above statement is weakened in the case of overlapping time windows as some customers inside $u$ may be restricted by their assigned time windows such that no consecutive block of customers can be formed. Therefore, a larger amount of time than $(II)$ may be required for tour $\mathcal{A}$ after the insertion of customer $j$.

The insertion of $j$ into $\mathcal{A}$ is infeasible if the following holds:

$\displaystyle\max\limits_{i\in\Theta(j,\mathcal{A})}~\lambda_{{u}}(\mathcal{A}+_{i}j)<0.$

(1)

We note that Condition (1) solely depends on the customers inside $u$. Hence, in case of non-overlapping time windows, it is only dependent on the customers assigned to time slot $u$.

Moreover, in case of non-overlapping time slots, the insertion of $j$ into $\mathcal{A}$ is feasible for at least one insertion position if the following inequality holds:

$\displaystyle\max\limits_{i\in\Theta(j,\mathcal{A})}~\lambda_{{u}}(\mathcal{A}+_{i}j)-\chi_{u}(\mathcal{A},j)\geq 0.$

(2)

Note that the statement does not hold in the case of overlapping time windows.

Finally, we can concisely describe the details of our ANS for the SOP as follows. We temporarily assign the new customer $j$ to each time window $u\in\mathcal{W}.$ For each tour $\mathcal{A}\in\mathcal{S}_{u}$ we apply the steps described below, which try to insert $j$ into $\mathcal{A}$ within its assigned time window $w(j)=u$. If this is possible, $u$ is added to the set of available time windows $\mathcal{T}_{j}\subseteq\mathcal{W}$.

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  Step 1 ensures that the current tour $\mathcal{A}$ fulfills CFEAS($j,\mathcal{A}$). . If $\sum\limits_{i\in[n]}c(i)+c(j)>C_{\mathcal{A}}$ holds, then customer $a$ cannot be feasibly inserted into tour $\mathcal{A}$. In this case, we have to reduce the overall weight $\sum\limits_{i\in[n]}c(i)$ of $\mathcal{A}$. This is achieved by applying local search operations 1move and 1swap, while as few operations as possible are used. Therefore, as much weight as possible is moved in each step and the operations stop once there is sufficient spare capacity on $\mathcal{A}$ to insert $j$, i.e., $\sum\limits_{i\in[n]}c(i)+c(j)\leq C_{\mathcal{A}}$ holds. If we do not succeed in modifying $\mathcal{A}$ such that CFEAS($j,\mathcal{A}$) holds, we terminate.

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  In Step 2 we aim to increase the free time $\lambda_{u}(\mathcal{A})$ through local search operations within Inside until the Infeasibility Condition (1) does not hold anymore. $\lambda_{u}(\mathcal{A})$ is increased by applying as few operations as possible. That way, the previously optimized schedule $\mathcal{S}_{u}$ is not altered more than necessary. If a local optimum is reached, meaning no further improvements can be achieved by local search operations, and (1) is still satisfied, the algorithm stops because the following steps cannot result in a feasible insertion of the new customer.

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  Step 3 is concerned with reducing the loss time $\chi_{u}(j,\mathcal{A})$ of time slot $u$ through local improvement operations within Outside. The operations are applied until either the new customer order can be inserted into the tour, the loss time is equal to zero $\chi_{u}(j,\mathcal{A})=0$, or a local optimum is reached.

• •

  Finally, in Step 4 we try to further increase the free time through local search operations within Inside. The free time is increased until either the insertion of customer $j$ is possible or a local optimum (of the free time objective) is reached and hence, we are not able to insert $j$ within time slot $u$.

Note that we apply local search operations during Steps 2-4 only if CFEAS($j,\mathcal{A}$) still holds. Further, we apply local search operations during Step 3 only if the Infeasibility Condition (1) does not hold for the resulting tour $\mathcal{A}$.

In this step, we aim to quickly identify all time windows $\mathcal{T}_{j}\subseteq\mathcal{W}$ during which a new customer $j$ can be inserted into (at least one of) the current tours. We compare to following approaches.

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  Simple Insertion heuristic,

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  ANS heuristic,

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  A feasibility version of the suggested MILPs for solving the TSPTW, which are applied for each time slot $u\in\mathcal{W}$ and for each $\mathcal{A}\in\mathcal{T}_{u}$.

Once customer $j$ has selected a time window $u\in\mathcal{T}_{j}$, we double-check its availability in the same way as in the Get TWs step and then immediately insert $j$ into $u$ at the best insertion point found. In contrast to the Get TWs step, we run the Simple Insertion or the ANS over all tours and all insertion points and select the feasible insertion point that results in the least increase of the total travel time of the schedule. Thus, we do not stop the algorithm after the first feasible insertion point is found. In the case of the TSPTW MILPs we apply their standard formulations for finding an optimal tour after the insertion of $j$, rather than their corresponding feasibility versions.

In this step we aim to reduce the total travel time of the schedule. We compare the following approaches.

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  1move + (1swap) improvement. The computationally cheap yet quite effective, Local Search heuristic builds the foundation of the Improvement step. We apply 1move  (and 1swap)   operations, where we focus on the 1move  operations if possible, because they are computationally cheaper and, in general, more effective than 1swap  operations. We stop our Local Search heuristic once we reach a local minimum of the objective function with respect to the selected neighborhood.

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  1move + 1swap + TSP(s)TW improvement. After applying the Local Search heuristic we additionally run our TSP(s)TW MILP on all tours that have changed since the last Improvement step. We use the current tours of our delivery schedule as the initial solution for TSP(s)TW MILP. While 1move and 1swap exchange customers between different tours, the TSP(s)TW MILP re-orders them within the tours, which makes it a useful complement to the Local Search heuristics. In practice, optimizing the single tours of a schedule to optimality has proven to be critical to ensure driver satisfaction as it guarantees that drivers do not encounter any inefficiencies on their routes.

The above improvement procedures are arranged in ascending order with respect to their computational effort.

During the Ordering Phase the proposed Local Search procedures only perform improving operations. However, the algorithms can be simply altered to a Simulated Annealing approach [20] by allowing also non-improving operations. However, this is more suitable for the Preparation Phase, when more time is available for improving the delivery schedule.

In order to allow a proper comparison of the methods for solving the SOP, we constructed instances which consist of

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  A feasible schedule which contains $p$ customers, and

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  a new customer order for which the availability of delivery time slots must be decided.

To create SOP instances for benchmarking we had to create feasible delivery schedules that are already filled with orders. Hence, we created delivery schedules by iteratively trying to insert $2000$ customers into each schedule. The Simple Insertion heuristic was used to conduct the feasibility checks. The number of customers that are contained in the resulting schedule is denoted by $\hat{p}$. We consider $\hat{p}$ being a sufficiently good approximation of the maximal number that can be inserted into a schedule considering a given configuration. Hence, we distinguish two scenarios.

1. 1.

   In the first scenario, we perform no optimization between the insertion steps.

2. 2.

   In the second scenario, the schedule is re-optimized by applying 1move after each customer insertion. This reduces the total travel time of the schedule. We restrict the Improvement step to the most simple approach to avoid unwanted bias when evaluating the Get TWs step.

In general, the schedules in the second scenario contain more orders while utilizing the same number of vehicles. Since the practical hardness of the SOP increases as the schedules get filled up with customers, we consider SOP instances with different fill levels. The fill level $f$ of a schedule is defined as the ratio between the number of customers $p$ in the schedule and the maximal number of customers $\hat{p}$ that can be inserted into the schedule. For benchmarking at a given fill level $f$ we select the schedule that was generated during above described process, containing $p=\lceil f\cdot\hat{p}\rceil$ orders. For each generated instance we solve the SOP using the Simple Insertion heuristic, the TSP(s)TW Insertion approach, and the proposed ANS heuristic. In order to investigate the differences between the considered methods, we analyze their performance on all three sets of time windows ($\operatorname*{\mathcal{W}_{\text{NO}}}$, $\operatorname*{\mathcal{W}_{\text{OV}1.5}}$, $\operatorname*{\mathcal{W}_{\text{OV}3}}$), instance sizes, and fill levels. Hence, we run tests on benchmark instances with $60$ tours (vehicles) and consider fill levels of $85\text{\,}\mathrm{\char 37\relax}$, $90\text{\,}\mathrm{\char 37\relax}$, $95\text{\,}\mathrm{\char 37\relax}$, and $99\text{\,}\mathrm{\char 37\relax}$.

We report the number of feasible time slots found by each method and required run times (mm:ss.zzz) for all scenarios, i.e., optimized and non-optimized schedules. We report the results for $\operatorname*{\mathcal{W}_{\text{NO}}}$ in Table 1, for $\operatorname*{\mathcal{W}_{\text{OV}1.5}}$ in Table 2, and for $\operatorname*{\mathcal{W}_{\text{OV}3}}$ in Table 3. Moreover, we report the number of feasible time slots that are found by combining the findings of all three methods, denoted as Combined in the tables. Additionally, we report $\hat{p}$, the number of customers at $100\text{\,}\mathrm{\char 37\relax}$ fill level. All reported numbers are average values over 100 instances each.

As reference to compare against, we select the Simple Insertion heuristic. Both other methods, TSP(s)TW Insertion and ANS, are entitled to find at least the delivery time slots that are determined by the Simple Insertion heuristic. This property is guaranteed due to the construction of those methods. Unfortunately, we can not provide an upper bound for the number of feasible delivery time slots as, to the best of our knowledge, there is no more powerful search method applicable for the SOP in the current literature. In our experiments, we restrict the ANS to 1move operations as preliminary experiments showed that allowing 1swap operations yields unacceptably long run times (up to some minutes). Also the results are only insignificantly better. Primarily, we notice that the Simple Insertion heuristic returns solutions for the SOP in less than one millisecond for all considered instances. At fill level $85\text{\,}\mathrm{\char 37\relax}$ the instances are still rather easy and hence, already the Simple Insertion heuristic determines nearly all time slots as being feasible. While the Simple Insertion heuristic still performs well at $90\text{\,}\mathrm{\char 37\relax}$ and $95\text{\,}\mathrm{\char 37\relax}$ on optimized schedules, it performs poorly on non-optimized schedules with the same fill levels due to the lower quality of the schedules.

Further, we observe that the TSP(s)TW Insertion approach yields a slight improvement over the Simple Insertion heuristic in terms of available time windows at $85\text{\,}\mathrm{\char 37\relax}$ – $95\text{\,}\mathrm{\char 37\relax}$. However, a significant improvement can be observed when it is applied to non-optimized schedules at $99\text{\,}\mathrm{\char 37\relax}$ fill level. TSP(s)TW Insertion shows acceptable run times for $\operatorname*{\mathcal{W}_{\text{NO}}}$. In contrast, run times for $\operatorname*{\mathcal{W}_{\text{OV}1.5}}$ and $\operatorname*{\mathcal{W}_{\text{OV}3}}$ are between $3$ seconds and nearly $4$ minutes and thus, unacceptable. Hence, considering these findings, the TSP(s)TW Insertion turns out to be impractical for AHD systems. Moreover, we notice that the ANS yields significantly more feasible time slots than Simple Insertion (and TSP(s)TW Insertion) on non-optimized schedules at $95\text{\,}\mathrm{\char 37\relax}$ and $99\text{\,}\mathrm{\char 37\relax}$. Similar behavior can be observed on optimized schedules at $99\text{\,}\mathrm{\char 37\relax}$. The run times of the ANS stay below $1$ second for nearly all experiments with up to $95\text{\,}\mathrm{\char 37\relax}$ fill level. The ANS clearly performs best in terms of solution quality on instances having $99\text{\,}\mathrm{\char 37\relax}$ fill level resulting in up to $11$ times more available delivery time windows than Simple Insertion. However, on those instances its run time reaches up to $9$ seconds. It is worth pointing out that the performance of the ANS is nearly constant over all three sets of time windows showing that it can also deal with instances having overlapping delivery time windows.

In general, we notice a slight performance drop of the ANS (compared to TSP(s)TW Insertion) from optimized schedules to non-optimized schedules. This can be explained by the fact that identifying feasible time windows is less hard for non-optimized schedules as they contain less orders on average. Also there is more potential for improvement when applying TSP(s)TW Insertion as the single tours have not been improved in any way after inserting the customers. Also, we observe that Combined shows only a marginal improvement over the ANS for optimized schedules. On the other hand, we notice a strong improvement of Combined compared to TSP(s)TW Insertion and ANS for non-optimized schedules at $99\text{\,}\mathrm{\char 37\relax}$ fill level. This can be explained by the fact that TSP(s)TW Insertion has a larger potential of finding a feasible insertion (compared to Simple Insertion) if the tours are of low quality (as there is more room to rearrange the tours) and the already observed performance drop of the ANS.

Our experiments show that the ANS heuristic is capable of finding a larger number of feasible delivery slots than the Simple Insertion heuristic, requiring run times that are suited for AHD services when dealing with moderately sized problem instances. However, to efficiently tackle very large instances, parallelization of the ANS is advised.

In summary, the ANS heuristic is clearly the best method for solving the SOP when being concerned with the solution quality while the Simple Insertion heuristic is the method of choice in case of very tight time restrictions.

First, we want to analyze the Improvement approaches for instance sizes which we found to appear most commonly in practice. Hence, we consider $500$ customers that are served by $16/18/20$ vehicles having capacity of $200$ units each. The number of used vehicles corresponds to the practical difficulty of the instances. The numbers are chosen such that the instances are reasonably difficult. In that sense, using $20$ vehicles results in accepting nearly all $500$ customers on average. These instances are designed to reflect the majority of delivery regions as they were encountered during our project with a leading supermarket chain.

The results for these experiments are reported in Table 4. We observe that all approaches considered are applicable in an online service as the average run time per step is below $4$ seconds, which is very reasonable for instances of this size. Further, a reduction of our objective function by $0.29\text{\,}\mathrm{\char 37\relax}$ to $0.60\text{\,}\mathrm{\char 37\relax}$ per step is remarkable as between two Improvement steps the schedule is altered only by the insertion of one customer. This can be further underlined by the reported average reduction of the cost of inserting the new customer ranging from $35.82\text{\,}\mathrm{\char 37\relax}$ to $63.48\text{\,}\mathrm{\char 37\relax}$. These numbers show that our approaches meet the requirements of modern AHD systems. The experiments reveal that 1move + 1swap clearly outperforms 1move in terms of improving the objective function (across all three types of time windows). Also solving the TSPTW afterwards results in further improvement of the objective. Considering different delivery time windows, we notice that the approaches perform best with respect to runtime on instances with $\operatorname*{\mathcal{W}_{\text{NO}}}$ and worst on instances with $\operatorname*{\mathcal{W}_{\text{OV}1.5}}$. While there is a slight difference for the Local Search operations, the difference is nearly 3 seconds when additionally applying the TSPTW MILP. This is due to the fact that the absence of overlapping time windows allows for a more efficient MILP formulation (Section 4.4). Thus, during peak times and in case of overlapping time windows we advise to stick to 1move  (or 1move + 1swap). Further, the average number of customers that can be inserted into the schedule deviates at most by $9$ (+$1.9\text{\,}\mathrm{\char 37\relax}$) between $\operatorname*{\mathcal{W}_{\text{NO}}}$ and $\operatorname*{\mathcal{W}_{\text{OV}1.5}}$. Hence, allowing overlapping time windows accounts for a small benefit concerning the degree of capacity utilization. Similarly, in case of overlapping time windows ($\operatorname*{\mathcal{W}_{\text{OV}1.5}}$ and $\operatorname*{\mathcal{W}_{\text{OV}3}}$) the travel time reduction is slightly larger than for $\operatorname*{\mathcal{W}_{\text{NO}}}$.

Large supermarket chains offer their services across the whole country. Caused by the different geographies, the sizes of the delivery regions that are covered by a depot strongly vary ranging from a few hundred up to around $2000$ customers per day. Dealing with such large delivery regions is especially challenging during periods of many customer request coming in within a short time frame. To accommodate these periods of high request frequency, we propose to run the Improvement step only after each $i$th successful Insertion step, instead of after each.

We want to validate this idea by running computational experiments. We consider instances with $2000$ customers (the largest number we encountered in practice) and $80$ vehicles for these experiments and report the results in Table 5. Each column shows results for different values of $i$. For these experiments we can only report the average improvement of the schedule when applying the respective improvement strategy. First, we notice that the runtime of the Improvement step increases with $i$: the more often we skip an Improvement step, the longer it takes to improve the schedule’s total travel time. Moreover, we observe an increased improvement per step with increasing $i$. Apparently, the improvement that was omitted can be made up (up to a certain extent), by applying the Improvement step at a later point in time. It shows that performing the Improvement step only after every $i$th successful insertion is a viable option. While 1move stays below $6$ seconds for $i=10$, its runtime increases up to $13$ seconds for $i=30$. The runtimes of 1move + 1swap are between $1$ minute and $2$ minutes (overlapping time windows), which is still acceptable. We observe that running the TSPTW MILP does increase the runtimes (on top of Local Search) only insignificantly while still showing some additional improvements of the objective. In general, the observations from the previous experiments with $500$ customers carry over to this experiments. Again we notice shorter runtimes for $\operatorname*{\mathcal{W}_{\text{NO}}}$ than for $\operatorname*{\mathcal{W}_{\text{OV}1.5}}$ and $\operatorname*{\mathcal{W}_{\text{OV}3}}$. However, we observe less significant differences than for the experiments with $500$ customers.

In summary, the results show that skipping the Improvement step allows us to deal with temporarily high customer request rates, even for very large schedules with a vast number of customers. Furthermore, we see that applying the Improvement step less often leads to an increased improvement per step at the cost of longer runtimes. Finally, note that triggering the Improvement step dynamically when there are no new requests is also a valid option.