Evaluation of the size of time windows for the travelling salesman problem in delivery operations

Abstract

A great challenge in operational research is to apply time-efficient algorithms to find the optimal solutions to the travelling salesman problem (TSP) and its many variations. The TSP with time windows (TSPTW) arises due to intense pressure for business to improve customer service. As online shopping becomes more popular, customer satisfaction increases if customers can decide when their orders are delivered to them. Customers may choose a time window, which is defined by an earliest delivery time and a latest delivery time, during which the package is delivered. Delivering packages to multiple customers is a typical TSPTW. One main challenge for a delivery business is to determine the size of the time window (i.e., the difference between the latest and earliest delivery times), which affects delivery cost and customer satisfaction. Although many previous studies investigated the TSPTW, none of those focused on the size of time windows. This study is the first that experiments with different time window sizes and determines their impact on tour duration, customer satisfaction, and solution time of the optimal delivery routes. The experiment results show that increasing the size of the time window decreases tour duration and customer satisfaction and increases solution time. Decreasing the size of the time window increases tour duration and customer satisfaction and decreases solution time. A small solution time is necessary for the scheduling of deliveries to many customers. A large solution time prevents a delivery business from delivering packages using optimal routes, which increases delivery cost and decreases customer satisfaction. The results of this study indicate that a general guideline for business is to allow customers to choose a time window size that is within the cost limit but is sufficiently small to maximize customer satisfaction and optimize delivery routes.

Introduction

Home delivery business such as DHL [29], FedEx [17], and UPS [33] guarantees delivery to customers on a chosen day. These industry leaders provide a delivery time window of one day. As the competition among home delivery business intensifies, companies compete in two fronts. First, companies try to deliver products to customers as fast as possible. For example, Amazon has been providing two-day deliveries for several years and has recently started one-day deliveries for selected products. This is a dramatic improvement for customer satisfaction since some other companies are still offering 7–10 days for deliveries. Secondly, companies try to let customers choose a narrower delivery window so that customers are present when packages are delivered. This is preferred by many customers and is especially important for high-value packages, packages require signatures (e.g., alcohol products), or customers in densely populated areas.

While companies have made strides in shortening the lead time between the order time and delivery time, there is limited progress in allowing customers to choose a reasonably small delivery time window [5, 26]. Due to unpredictable traffic and conditions of the delivery vehicle, driver performance, and other factors, companies are reluctant to commit to small delivery time windows. If a delivery is not completed in the specified time window, the company may incur financial loss and subject to other consequences such as lawsuits. The risk of providing customers with small delivery time windows can be factored into the delivery cost. The fundamental challenge in determining the size of the delivery time window is how to quantify the tradeoff between delivery cost and customer satisfaction when the size of time windows changes. Managers and practitioners intuitively understand that a small-time window increases delivery cost and customer satisfaction. There is a strong need to determine how the delivery cost and customer satisfaction change as the size of the delivery time window changes. This article answers this important question through mathematical modeling and case studies.

This article applies an innovative mathematical model [15] that obtains the optimal delivery route within seconds for a typical home package delivery problem with approximately 40 customers. To the best of the authors’ knowledge, this research is the first study that conducts sensitivity analysis and investigates how the delivery cost and customer satisfaction vary as the size of the delivery time window changes. Results show that smaller time windows increase delivery cost but improve customer satisfaction whereas larger time windows decrease the delivery cost but negatively affect customer satisfaction. There is a range for the size of the delivery time windows that balances customer satisfaction and delivery cost.

Background

Recent development in the travelling salesman problem with time windows (TSPTW) focused on four areas: mathematical modeling, travel times, time windows, and electric or hybrid vehicles. Yuan et al. [32] proposed lifted versions of the subtour elimination constraints. Papalitsas et al. [28] formulated the TSPTW as a quadratic unconstrained binary optimization problem and used quantum computing to solve the model. These mathematical modeling tools continue to extend the applications of the TSPTW. Since the travel time between nodes is stochastic rather than deterministic in many applications, several studies expanded the TSPTW to include varying travel times. Montero et al. [25] used integer linear programming to model the time-dependent TSPTW, where the travel time depends on the travel speed, and developed an exact algorithm to solve the model. Arigliano et al. [2] also investigated and solved the time-dependent TSPTW using the branch-and-bound algorithm.

Several studies focused on time windows in the TSPTW. For example, Fachini and Armentano [11] proposed exact and heuristic dynamic programming algorithms for the TSPTW in which the size of time windows may be increased. In other words, the service of a customer may start before the earliest service time or complete after the latest service time with a penalty cost. Similarly, Avraham and Raviv [3] studied the TSPTW with soft time windows and introduced a specialized branch-and-bound algorithm and an adaptive large neighborhood search heuristic for the problem. The size of time windows in these studies varies from a few time units to a few hundred time units. These studies did not determine how delivery cost or customer satisfaction may be affected by the size of time windows. Instead, these studies assumed that the size of time windows could vary and focused on various algorithms that solved the TSPTW.

The package delivery problem studied in this article is related to the general vehicle routing problem. In both the package delivery and vehicle routing problems, the optimal route of a vehicle is identified to minimize either the total travel distance or travel time of the vehicle. The vehicle routing problem has been studied since 1960s [4, 6, 8, 9, 12, 14, 18,19,20,21, 27]. Haghani and Jung [13] presented a genetic algorithm to solve a pick-up or delivery vehicle routing with soft time windows. The study considered multiple vehicles with different capacities, real-time service requests, and dynamic travel times between destinations. Almoustafa et al. [1] improved a branch-and-bound method to solve the asymmetric distance-constrained vehicle routing suggested by Laporte et al. [20]. Chen et al. [7] formulated a real-time time-dependent vehicle routing with time windows as a series of mixed-integer programming models and developed a heuristic algorithm, which included route construction and improvement. Spliet and Gabor [31] proposed a formulation of a time window asymmetric vehicle routing and developed two variants of a column generation algorithm to solve the linear programming relaxation of this formulation. Kritzinger et al. [16] applied a variable neighborhood search algorithm to solve the time-dependent vehicle routing with time windows.

Mathematical model and experiments

Package delivery business face multi-facet challenges in profitability and customer satisfaction. For example, the Chinese e-commerce giant, JD.com, makes 90% of Chinese deliveries within 24 h and 57% of their deliveries arrive within 12 h [22]. JD.com has forgone most profit to build up its nationwide logistics system, including 65,000 staff couriers who deliver on bicycles and in small vans in China. A key component determining the cost segmentation and profit margin is the total travel distance or tour duration (total delivery time) of a delivery vehicle. A shorter travel distance or tour duration decreases the cost and increases the profit margin. On the other hand, customer satisfaction in package delivery is affected by the lead time between issuing and receiving the order, choices of delivery (e.g., time windows for delivery), and other subtle and underlying factors such as packaging of the order and greetings from the delivery personnel.

The profitability and customer satisfaction are often conflicting objectives. A longer lead time helps increase profitability by reducing the inventory and order processing cost. A delivery company also prefers to deliver packages during time periods with less traffic, and group customers that are geographically close for deliveries during the same time period. All these cost-reduction practices negatively affect customer satisfaction. There are several ways to improve customer satisfaction. For example, a short lead time greatly enhances customer satisfaction and increases a company’s competitiveness, but negatively impacts the company’s profitability [22]. For another example, allowing customers to choose a delivery time window during which the delivery is made greatly improves customer satisfaction but unavoidably increases cost. This study investigates how the size of the time window affects the tradeoff between cost and customer satisfaction.

The cost is determined by the tour duration and solution time. The tour duration is the total delivery time. The solution time is the time it takes to solve the mathematical model and identify the optimal delivery route. The cost increases as the tour duration or solution time increases and decreases as the tour duration or solution time decreases. Customer satisfaction is determined by the size of the delivery time window. Customer satisfaction increases when the size of the delivery time window decreases and decreases when the size of the delivery time window increases.

The package delivery problem studied in this research is a TSPTW. This TSPTW is modeled as an integer linear programming model (Table 1; [15]). The model is applied to a set of benchmark instances [10], https://homepages.dcc.ufmg.br/~rfsilva/tsptw/) and the General Algebraic Modeling System (GAMS) is used to solve the instances and find the optimal routes that minimize the tour duration. The size of delivery time windows in the benchmark problems is systematically adjusted to determine its impact on the tradeoff between cost and customer satisfaction. The mathematical model [15] adopted in this research requires the least amount of computation time to identify the optimal routes and is readily available for adjusting delivery time windows.

Table 1 Mathematical model for the TSPTW [15]

Equation (1) in Table 1 is the objective function of the mathematical model and aims to minimize the tour duration of a package delivery vehicle. Equations (2)–(8) are constraints. Equation (2) initiates the arrival time of the vehicle at the first node (customer). Equations (3) and (7) define a time window for each node within which the vehicle may arrive and deliver the package. Equation (4) is an innovative step-by-step sub-tour elimination constraint. When \({x}_{ij}=1\), indicating that the vehicle travels from \(i\) to \(j\), Eq. (4) becomes \({t}_{ij}\le {t}_{j}-{t}_{i}\) and ensures that the difference between the arrival times at \(i\) and \(j\) is at least the travel time from \(i\) to \(j\), When \({x}_{ij}=0\), indicating that the vehicle does not travel from \(i\) to \(j\), Eq. (4) becomes \({t}_{i}-{t}_{j}\le {b}_{i}-{a}_{j}\). There are two possible scenarios: the vehicle arrives at \(i\) before it arrives at \(j\), or the vehicle arrives at \(j\) before it arrives at \(i\). If the vehicle arrives at \(i\) before it arrives at \(j\), Eq. (4) can be rewritten as \({t}_{j}-{t}_{i}\ge {a}_{j}-{b}_{i}\). Since the vehicle arrives at \(j\) after it arrives at \(i\), \({a}_{j}-{b}_{i}\) is the minimum difference between \({t}_{j}\) and \({t}_{i}\), Eq. (4) always holds. If the vehicle arrives at \(j\) before it arrives at \(i\), \({b}_{i}-{a}_{j}\) in Eq. (4) is the maximum difference between \({t}_{i}\) and \({t}_{j}\), and Eq. (4) always holds. Equations (5) and (6) ensure that the delivery vehicle visits each node (customer) once and only once. Equation (8) determines the tour duration.

The main purpose of this study is to investigate how the size of time windows affects tour duration, customer satisfaction, and solution time of the mathematical model. The size of delivery time windows may vary in many different ways. The experiments in this study use the problems in the TSPTW library [30] and systematically change the size of time windows. Each problem in the TSPTW library includes multiple time windows, each of which is defined by an earliest time and a latest time within which the delivery must be made to the customer. This study adjusts the size of time windows using three approaches, proportion, normalization, and stepwise. The proportion approach multiplies the size of each time window by the same coefficient. The difference between the sizes is magnified (with a coefficient greater than one) or diminished (with a coefficient less than one). The normalization approach ensures that all time windows have the same size. The difference between the sizes become zero. The stepwise approach takes a middle-of-the-road path and increases or decreases the size of each time window by two units (one unit for each side of a time window) at a time. The difference between the sizes remain the same.

In the proportion approach, the size of a time window is increased or decreased by multiplying a coefficient between “0” and “∞.” The earliest and latest times of a time window are adjusted by the same amount to produce the change in the size of the time window. Figure 1 illustrates the proportion approach. \({S}_{i}\) is the size of a time window \(i\) in the TSPTW library [30]. \({a}_{i}\) is the earliest delivery time of the time window \(i\) and \({b}_{i}\) is the latest delivery time of the time window \(i\). \({S}_{i}={b}_{i}-{a}_{i}\). The experiments multiply six coefficients, 0, 0.5, 1, 2, 10, and ∞, and \({S}_{i}\) to produce six time windows, 0, \(0.5{S}_{i}\), \({S}_{i}\), 2 \({S}_{i}\), 10 \({S}_{i}\), and ∞, as represented by six orange bars from left to right in Fig. 1, respectively. Each time window’s earliest and latest delivery times, \({a}_{i}^{*}\) and \({b}_{i}^{*}\), are obtained by adjusting \({a}_{i}\) and \({b}_{i}\), respectively, by the same amount. For example, to produce a time window with a size of \(0.5{S}_{i}\), \({a}_{i}^{*}\) of the time window is \({a}_{i}+0.25{S}_{i}\) and \({b}_{i}^{*}\) of the time window is \({b}_{i}-0.25{S}_{i}\). The size of the time window is, therefore, \({b}_{i}^{*}-{a}_{i}^{*}=\left({b}_{i}-0.25{S}_{i}\right)-\left({a}_{i}+0.25{S}_{i}\right)=\left({b}_{i}-{a}_{i}\right)-0.5{S}_{i}={S}_{i}-0.5{S}_{i}=0.5{S}_{i}\).

Fig. 1
figure1

Proportional adjustment of time windows

The normalization approach adjusts time windows to ensure that all time windows have the same size. The experiments use two different ways to produce the same size for all time windows: equal maximum size and equal minimum size. Table 2 illustrates the normalization approach. Suppose there are five-time windows in a benchmark problem in the TSPTW library. The maximum size of these five-time windows is 60, which is the size of time window \(i=5\). The minimum size of these five-time windows is 10, which is the size of time window \(i=1\). In the experiments with equal maximum size, the size of all five-time windows is adjusted and is equal to 60. As in the proportion approach, both the earliest delivery time \({a}_{i}\) and the latest delivery time \({b}_{i}\) are adjusted by the same amount in the normalization approach. One exception is for time window \(i=1\). Since \({a}_{1}=0\) and cannot decrease further, \({b}_{1}\) increases by 50. In the experiments with equal minimum size, the size of all five-time windows is adjusted and is equal to 10, and both \({a}_{i}\) and \({b}_{i}\) are adjusted by the same amount.

Table 2 Example of normalization of time windows

The stepwise approach decreases \({a}_{i}\) and increases \({b}_{i}\) by one time unit at the same time. This incremental adjustment of one unit continues until the size of the time window is sufficiently large and equivalent to infinity. The stepwise approach captures the granularity of how changes in the size of a time window affects cost and customer satisfaction.

Results and discussion

The experiments implement the mathematical model (Table 1) in GAMS and uses five benchmark problems (n20.w20.0001, n20.w20.0002, n20.w20.0003, n20.w20.0004, and n20.w20.0005) in the TSPTW library (Silva and Urrutia, 2012) to find the optimal delivery routes that minimize the tour duration. The size of time windows in the experiments is adjusted according to the three approaches outlined in Sect. 3, proportion, normalization, and stepwise. The mathematical model is solved using GAMS win64 24.0.2 on a computer with Intel i7 CPU 870 @ 2.93 GHz, 12.0 GB RAM, and Windows 10 Enterprise. To illustrate how the size of time windows is adjusted, Appendices 1, 2, and 3 show adjusted time windows for the benchmark problem n20.w20.0001 according to the proportion, normalization, and stepwise approaches, respectively. The maximum allowed solution time for GAMS is set to one hour. If it takes more than one hour to solve the mathematical model and find the optimal routes, GAMS terminates after one hour and provides the best delivery route and minimum tour duration up to that point. Table 3 shows the experiment results of the proportion approach.

Table 3 Experiment results of time windows adjusted using the proportion approach

Table 3 depicts solution times (seconds) of finding the optimal delivery routes and tour durations (seconds) of optimal delivery routes for five benchmark problems in the TSPTW library (Silva and Urrutia, 2012). The size of time windows is adjusted proportionally. There are 17 different sizes from “0” to “∞.” When the size is “∞,” there is no requirement for a time window within which the delivery must be completed; packages may be delivered to a customer at any time. The tour duration with “Inf.” indicates that feasible delivery routes do not exist and the model is infeasible. In other words, no delivery routes can satisfy all delivery time windows. Dumas et al. [10] prepared these five benchmark problems and their feasible solution space is relatively small. When the size of time windows \({S}_{i}\)’s is reduced, the feasible region becomes smaller and the model may become infeasible.

The highlighted (yellow) column with the size \({S}_{i}\) in Table 3 includes experiment results for the original time windows \({S}_{i}\)’s in the five benchmark problems. As the size of delivery time windows increases, both tour duration and customer satisfaction decrease, and the solution time increases. Decreasing customer satisfaction is due to the increasing size of time windows. Smaller tour duration generally incurs less delivery cost. Larger solution time increases delivery cost. Results in Table 3 show that there is a tradeoff between the delivery cost and customer satisfaction. Larger time windows decrease the delivery cost and customer satisfaction. When the size of the time window is too large, however, the delivery cost will not decrease further and may increase because the solution time is too large and the optimal delivery route is not obtained. Smaller time windows increase the delivery cost and customer satisfaction. When the size of the time window is too small, however, customer satisfaction decreases because feasible delivery routes do not exist.

Table 3 reveals that the ideal size of time windows is between \({S}_{i}\) and \(3{S}_{i}\). When the size is larger than \(3{S}_{i}\), customer satisfaction deteriorates and the delivery cost may increase because the optimal delivery route is not obtained. When the size is smaller than \({S}_{i}\), feasible delivery routes may not exist. When the size is between \({S}_{i}\) and \(3{S}_{i}\), the solution time is at most a little over one second and its impact on the delivery cost is negligible. To reduce the delivery cost, larger time windows should be used for delivery operations. To increase customer satisfaction, smaller time windows should be made available to customers.

The highlighted (yellow) column with the size \({S}_{i}\) in Table 4 shows the same results as those in the highlighted (yellow) column in Table 3 for the original time windows \({S}_{i}\)’s in the five benchmark problems. When the size of all time windows is the same as the minimum size, none of the problems has any feasible solution and solution time is relatively small. When the size of all time windows is the same as the maximum size, all five problems are feasible and the minimum tour duration is identified. The solution time is larger and the largest solution time is around one second. Comparing the results for the original time windows \({S}_{i}\)’s and time windows with the same maximum size, the latter has worse customer satisfaction but does not decrease tour duration significantly. The maximum decrease in tour duration is 10 s (= 407–397) or about 2.5%. The tradeoff between delivery cost (tour duration and solution time) and customer satisfaction (size of time windows) is not clear for the normalization approach. When the size of time windows increases, the tour duration only decreases slightly.

Table 4 Experiment results of time windows adjusted using the normalization approach

The stepwise approach increases the size of the time windows gradually with the same amount of adjustment at each step. Table 5 shows the experiment results of the stepwise approach for problem n20.w20.0001 in the TSPTW library [30]. In the first experiment in Table 5, the original time windows \({S}_{i}\)’s in problem n20.w20.0001 are used to find the minimum tour duration and the solution time. The results for the original time windows \({S}_{i}\)’s are highlighted in yellow in Table 5. These are the same as those for \({S}_{i}\)’s in Tables 3 and 4. In each experiment that follows, the size of time windows increases by two seconds; this is achieved by decreasing \({a}_{i}\), the earliest delivery time, and increasing \({b}_{i}\), the latest delivery time, by one second at the same time. The tour duration and solution time of each experiment are included in Table 5.

Table 5 Experiment results of time windows adjusted using the stepwise approach for problem n20.w20.0001

Figure 2 is a histogram that visualizes tour durations in Table 5. Figure 2 clearly shows that the minimum tour duration decreases as the size of time windows increases. This is mainly because a larger time window leads to a larger feasible region, which in turn results in a better optimal solution, i.e., a smaller tour duration. Figure 3 shows how the solution time in Table 5 changes as the size of time windows increases. The solution time remains small, around a few second or less, until the time window reaches \({S}_{i}\)+90, which requires a solution time of about 744 s. For time windows that are larger than \({S}_{i}\)+90, the solution time varies but mostly remains relatively large. Table 5 and Fig. 3 show that large solution times can occur when time windows are large enough.

Fig. 2
figure2

Tour durations of the stepwise approach summarized in Table 5

Fig. 3
figure3

Solution times of the stepwise approach summarized in Table 5

The stepwise approach also reveals the tradeoff between customer satisfaction and delivery cost. As the size of time windows increases (Fig. 2), customer satisfaction decreases while the tour duration decreases and solution time remains stable, indicating reduced delivery cost. There is a caveat when the time windows become too large. Figure 3 shows that the solution time of identifying the minimum tour duration increases dramatically when the size of time windows reaches \({S}_{i}\)+90. In other words, a common and intuitive practice by many delivery companies to adopt large delivery time windows leads to both poor customer satisfaction and high cost (the optimal route is not obtained due to large solution time). While certain items may be delivered without customers being present, poor customer satisfaction resulted from large delivery time windows is exacerbated by the fact that sometimes customers are forced to wait at home for deliveries (e.g., alcohol deliveries, weather conditions, and requests by the senders).

The experiment results of the three approaches, proportion, normalization, and stepwise, provide important guidelines for determining time windows in-home delivery operations. First, customers of many delivery operations can choose the best time for a delivery to be made. A delivery company should provide customers with an appropriate size of the delivery time window to balance customer satisfaction and cost of delivery. This is a multi-objective optimization problem. Secondly, there is a tradeoff between customer satisfaction and delivery cost (Tables 3 and 5). Better customer satisfaction (smaller delivery time windows) require a higher delivery cost (larger tour duration). Thirdly, when the size of time windows is too small, it becomes infeasible to deliver to multiple customers and satisfy all delivery time windows (Tables 3 and 4). On the other hand, when the size of time windows is too large, the time it takes to find the minimum tour duration increases significantly (Table 3 and 5). A delivery company should avoid delivery time windows that are too small or too large.

Conclusions

This study investigates the impact of the size of delivery time windows on customer satisfaction and delivery cost. The results of this study suggest that a delivery company should not use delivery time windows that are either too small or too large. Extremely small time windows render the delivery operations infeasible; multiple deliveries cannot be completed to satisfy narrow time windows. Extremely large time windows not only lead to poor customer satisfaction but also require a significant amount of solution time to find the minimum tour duration, which is not obtained and, therefore, increases the delivery cost. Conventional wisdom suggests that large time windows reduce the delivery cost because delivery companies have more flexibility in choosing delivery routes with large delivery time windows. The results of this study show that it becomes practically infeasible (solution time exceeds one hour) to find the delivery route that minimizes the tour duration when time windows are too large. Extremely large time windows result in poor customer satisfaction and high delivery cost and require significant solution time for route planning.

This study also suggests that there is a tradeoff between customer satisfaction and delivery cost. As the size of time windows increases, both customer satisfaction and delivery cost decrease. The latter is due to smaller tour duration resulted from larger time windows. Future research may determine the most appropriate size of delivery time windows for a variety of companies that deliver packages to homes and businesses. One approach is to develop multi-objective optimization models that take into consideration of multiple objectives such as customer satisfaction, tour duration, and solution time, and various constraints such as road traffic, municipality ordinances and codes, and labor standards.

This study experiments with five benchmark problems in the TSPTW library [30], which also includes other benchmark problems. There are many more TSPTW benchmark problems that are available in the public domain (e.g., [23, 24]. Another important future research direction is to expand this study and conduct experiments on additional benchmark problems. These additional experiments are expected to validate the conclusions obtained in this study and may provide more insight into the tradeoff between customer satisfaction and cost when the size of time windows is adjusted in the home delivery business.

Availability of data and material

All data are included in the article.

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Appendices

Appendix 1: Adjusted time windows of the benchmark problem n20.w20.0001 using the proportion approach

\(i\) 0 \(0.2{S}_{i}\) \(0.4{S}_{i}\) \(0.6{S}_{i}\) \(0.8{S}_{i}\) \({S}_{i}\) \(1.2{S}_{i}\) \(1.4{S}_{i}\)
\({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}\) \({b}_{i}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\)
1 0 408 0 408 0 408 0 408 0 408 0 408 0 448.8 0 489.6
2 65 65 64.4 65.6 63.8 66.2 63.2 66.8 62.6 67.4 62 68 61.4 68.6 60.8 69.2
3 193 193 190.6 195.4 188.2 197.8 185.8 200.2 183.4 202.6 181 205 178.6 207.4 176.2 209.8
4 315 315 313.2 316.8 311.4 318.6 309.6 320.4 307.8 322.2 306 324 304.2 325.8 302.4 327.6
5 215.5 215.5 215.2 215.8 214.9 216.1 214.6 216.4 214.3 216.7 214 217 213.7 217.3 213.4 217.6
6 56 56 55 57 54 58 53 59 52 60 51 61 50 62 49 63
7 115.5 115.5 112.8 118.2 110.1 120.9 107.4 123.6 104.7 126.3 102 129 99.3 131.7 96.6 134.4
8 180.5 180.5 179.4 181.6 178.3 182.7 177.2 183.8 176.1 184.9 175 186 173.9 187.1 172.8 188.2
9 256.5 256.5 255.2 257.8 253.9 259.1 252.6 260.4 251.3 261.7 250 263 248.7 264.3 247.4 265.6
10 13 13 11 15 9 17 7 19 5 21 3 23 1 25 0 27
11 35 35 32.2 37.8 29.4 40.6 26.6 43.4 23.8 46.2 21 49 18.2 51.8 15.4 54.6
12 84.5 84.5 83.4 85.6 82.3 86.7 81.2 87.8 80.1 88.9 79 90 77.9 91.1 76.8 92.2
13 87 87 85.2 88.8 83.4 90.6 81.6 92.4 79.8 94.2 78 96 76.2 97.8 74.4 99.6
14 147 147 145.6 148.4 144.2 149.8 142.8 151.2 141.4 152.6 140 154 138.6 155.4 137.2 156.8
15 370 370 366.8 373.2 363.6 376.4 360.4 379.6 357.2 382.8 354 386 350.8 389.2 347.6 392.4
16 52.5 52.5 50.4 54.6 48.3 56.7 46.2 58.8 44.1 60.9 42 63 39.9 65.1 37.8 67.2
17 7.5 7.5 6.4 8.6 5.3 9.7 4.2 10.8 3.1 11.9 2 13 0.9 14.1 0 15.2
18 33 33 31.2 34.8 29.4 36.6 27.6 38.4 25.8 40.2 24 42 22.2 43.8 20.4 45.6
19 26.5 26.5 25.2 27.8 23.9 29.1 22.6 30.4 21.3 31.7 20 33 18.7 34.3 17.4 35.6
20 15 15 13.8 16.2 12.6 17.4 11.4 18.6 10.2 19.8 9 21 7.8 22.2 6.6 23.4
21 287.5 287.5 285 290 282.5 292.5 280 295 277.5 297.5 275 300 272.5 302.5 270 305
22 0 408 0 408 0 408 0 408 0 408 0 408 0 448.8 0 489.6
\(i\) \(1.6{S}_{i}\) \(1.8{S}_{i}\) \(2{S}_{i}\) \(3{S}_{i}\) \(4{S}_{i}\) \(5{S}_{i}\) \(10{S}_{i}\) \(20{S}_{i}\) \(\infty \)
\({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\)
1 0 530.4 0 571.2 0 612 0 816 0 1020 0 1224 0 2244 0 4284 0 41,208
2 60.2 69.8 59.6 70.4 59 71 56 74 53 77 50 80 35 95 5 125 0 668
3 173.8 212.2 171.4 214.6 169 217 157 229 145 241 133 253 73 313 0 433 0 2605
4 300.6 329.4 298.8 331.2 297 333 288 342 279 351 270 360 225 405 135 495 0 2124
5 213.1 217.9 212.8 218.2 212.5 218.5 211 220 209.5 221.5 208 223 200.5 230.5 185.5 245.5 0 517
6 48 64 47 65 46 66 41 71 36 76 31 81 6 106 0 156 0 1061
7 93.9 137.1 91.2 139.8 88.5 142.5 75 156 61.5 169.5 48 183 0 250.5 0 385.5 0 2829
8 171.7 189.3 170.6 190.4 169.5 191.5 164 197 158.5 202.5 153 208 125.5 235.5 70.5 290.5 0 1286
9 246.1 266.9 244.8 268.2 243.5 269.5 237 276 230.5 282.5 224 289 191.5 321.5 126.5 386.5 0 1563
10 0 29 0 31 0 33 0 43 0 53 0 63 0 113 0 213 0 2023
11 12.6 57.4 9.8 60.2 7 63 0 77 0 91 0 105 0 175 0 315 0 2849
12 75.7 93.3 74.6 94.4 73.5 95.5 68 101 62.5 106.5 57 112 29.5 139.5 0 194.5 0 1190
13 72.6 101.4 70.8 103.2 69 105 60 114 51 123 42 132 0 177 0 267 0 1896
14 135.8 158.2 134.4 159.6 133 161 126 168 119 175 112 182 77 217 7 287 0 1554
15 344.4 395.6 341.2 398.8 338 402 322 418 306 434 290 450 210 530 50 690 0 3586
16 35.7 69.3 33.6 71.4 31.5 73.5 21 84 10.5 94.5 0 105 0 157.5 0 262.5 0 2163
17 0 16.3 0 17.4 0 18.5 0 24 0 29.5 0 35 0 62.5 0 117.5 0 1113
18 18.6 47.4 16.8 49.2 15 51 6 60 0 69 0 78 0 123 0 213 0 1842
19 16.1 36.9 14.8 38.2 13.5 39.5 7 46 0.5 52.5 0 59 0 91.5 0 156.5 0 1333
20 5.4 24.6 4.2 25.8 3 27 0 33 0 39 0 45 0 75 0 135 0 1221
21 267.5 307.5 265 310 262.5 312.5 250 325 237.5 337.5 225 350 162.5 412.5 37.5 537.5 0 2800
22 0 530.4 0 571.2 0 612 0 816 0 1020 0 1224 0 2244 0 4284 0 41,208

Appendix 2: Adjusted time windows of the benchmark problem n20.w20.0001 using the normalization approach

\(i\) Time windows in the TSPTW library Adjusted time windows with equal minimum size Adjusted time windows with equal maximum size
\({a}_{i}\) \({b}_{i}\) \({S}_{i}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({S}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({S}_{i}^{*}\)
1 0 408 408 0 408 408 0 408 408
2 62 68 4 63.5 66.5 3 49 81 32
3 181 205 24 191.5 194.5 3 177 209 32
4 306 324 18 313.5 316.5 3 299 331 32
5 214 217 3 214 217 3 199.5 231.5 32
6 51 61 10 54.5 57.5 3 40 72 32
7 102 129 27 114 117 3 99.5 131.5 32
8 175 186 11 179 182 3 164.5 196.5 32
9 250 263 13 255 258 3 240.5 272.5 32
10 3 23 20 11.4 14.5 3 0 32 32
11 21 49 28 33.5 36.5 3 19 51 32
12 79 90 11 83 86 3 68.5 100.5 32
13 78 96 18 85.5 88.5 3 71 103 32
14 140 154 14 145.5 148.5 3 131 163 32
15 354 386 32 368.5 371.5 3 354 386 32
16 42 63 21 51 54 3 36.5 68.5 32
17 2 13 11 6 9 3 0 32 32
18 24 42 18 31.5 34.5 3 17 49 32
19 20 33 13 25 28 3 10.5 42.5 32
20 9 21 12 13.5 16.5 3 0 32 32
21 275 300 25 286 289 3 271.5 303.5 32
22 0 408 408 0 408 408 0 408 408

Appendix 3: Adjusted time windows of the benchmark problem n20.w20.0001 using the stepwise approach

\(i\) \({S}_{i}\) \({S}_{i}+2\) \({S}_{i}+4\) \({S}_{i}+6\) \({S}_{i}+8\) \({S}_{i}+10\) \({S}_{i}+12\) \({S}_{i}+14\)
\({a}_{i}\) \({b}_{i}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\)
1 0 408 0 409 0 410 0 411 0 412 0 413 0 414 0 415
2 62 68 61 69 60 70 59 71 58 72 57 73 56 74 55 75
3 181 205 180 206 179 207 178 208 177 209 176 210 175 211 174 212
4 306 324 305 325 304 326 303 327 302 328 301 329 300 330 299 331
5 214 217 213 218 212 219 211 220 210 221 209 222 208 223 207 224
6 51 61 50 62 49 63 48 64 47 65 46 66 45 67 44 68
7 102 129 101 130 100 131 99 132 98 133 97 134 96 135 95 136
8 175 186 174 187 173 188 172 189 171 190 170 191 169 192 168 193
9 250 263 249 264 248 265 247 266 246 267 245 268 244 269 243 270
10 3 23 2 24 1 25 0 26 0 27 0 28 0 29 0 30
11 21 49 20 50 19 51 18 52 17 53 16 54 15 55 14 56
12 79 90 78 91 77 92 76 93 75 94 74 95 73 96 72 97
13 78 96 77 97 76 98 75 99 74 100 73 101 72 102 71 103
14 140 154 139 155 138 156 137 157 136 158 135 159 134 160 133 161
15 354 386 353 387 352 388 351 389 350 390 349 391 348 392 347 393
16 42 63 41 64 40 65 39 66 38 67 37 68 36 69 35 70
17 2 13 1 14 0 15 0 16 0 17 0 18 0 19 0 20
18 24 42 23 43 22 44 21 45 20 46 19 47 18 48 17 49
19 20 33 19 34 18 35 17 36 16 37 15 38 14 39 13 40
20 9 21 8 22 7 23 6 24 5 25 4 26 3 27 2 28
21 275 300 274 301 273 302 272 303 271 304 270 305 269 306 268 307
22 0 408 0 409 0 410 0 411 0 412 0 413 0 414 0 415
\(i\) \({S}_{i}+16\) \({S}_{i}+18\) \({S}_{i}+20\) \({S}_{i}+22\) \({S}_{i}+24\) \({S}_{i}+26\) \({S}_{i}+28\) \({S}_{i}+30\)
\({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\)
1 0 416 0 417 0 418 0 419 0 420 0 421 0 422 0 423
2 54 76 53 77 52 78 51 79 50 80 49 81 48 82 47 83
3 173 213 172 214 171 215 170 216 169 217 168 218 167 219 166 220
4 298 332 297 333 296 334 295 335 294 336 293 337 292 338 291 339
5 206 225 205 226 204 227 203 228 202 229 201 230 200 231 199 232
6 43 69 42 70 41 71 40 72 39 73 38 74 37 75 36 76
7 94 137 93 138 92 139 91 140 90 141 89 142 88 143 87 144
8 167 194 166 195 165 196 164 197 163 198 162 199 161 200 160 201
9 242 271 241 272 240 273 239 274 238 275 237 276 236 277 235 278
10 0 31 0 32 0 33 0 34 0 35 0 36 0 37 0 38
11 13 57 12 58 11 59 10 60 9 61 8 62 7 63 6 64
12 71 98 70 99 69 100 68 101 67 102 66 103 65 104 64 105
13 70 104 69 105 68 106 67 107 66 108 65 109 64 110 63 111
14 132 162 131 163 130 164 129 165 128 166 127 167 126 168 125 169
15 346 394 345 395 344 396 343 397 342 398 341 399 340 400 339 401
16 34 71 33 72 32 73 31 74 30 75 29 76 28 77 27 78
17 0 21 0 22 0 23 0 24 0 25 0 26 0 27 0 28
18 16 50 15 51 14 52 13 53 12 54 11 55 10 56 9 57
19 12 41 11 42 10 43 9 44 8 45 7 46 6 47 5 48
20 1 29 0 30 0 31 0 32 0 33 0 34 0 35 0 36
21 267 308 266 309 265 310 264 311 263 312 262 313 261 314 260 315
22 0 416 0 417 0 418 0 419 0 420 0 421 0 422 0 423
\(i\) \({S}_{i}+32\) \({S}_{i}+34\) \({S}_{i}+36\) \({S}_{i}+38\) \({S}_{i}+40\) \({S}_{i}+42\) \({S}_{i}+44\) \({S}_{i}+46\)
\({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\)
1 0 424 0 425 0 426 0 427 0 428 0 429 0 430 0 431
2 46 84 45 85 44 86 43 87 42 88 41 89 40 90 39 91
3 165 221 164 222 163 223 162 224 161 225 160 226 159 227 158 228
4 290 340 289 341 288 342 287 343 286 344 285 345 284 346 283 347
5 198 233 197 234 196 235 195 236 194 237 193 238 192 239 191 240
6 35 77 34 78 33 79 32 80 31 81 30 82 29 83 28 84
7 86 145 85 146 84 147 83 148 82 149 81 150 80 151 79 152
8 159 202 158 203 157 204 156 205 155 206 154 207 153 208 152 209
9 234 279 233 280 232 281 231 282 230 283 229 284 228 285 227 286
10 0 39 0 40 0 41 0 42 0 43 0 44 0 45 0 46
11 5 65 4 66 3 67 2 68 1 69 0 70 0 71 0 72
12 63 106 62 107 61 108 60 109 59 110 58 111 57 112 56 113
13 62 112 61 113 60 114 59 115 58 116 57 117 56 118 55 119
14 124 170 123 171 122 172 121 173 120 174 119 175 118 176 117 177
15 338 402 337 403 336 404 335 405 334 406 333 407 332 408 331 409
16 26 79 25 80 24 81 23 82 22 83 21 84 20 85 19 86
17 0 29 0 30 0 31 0 32 0 33 0 34 0 35 0 36
18 8 58 7 59 6 60 5 61 4 62 3 63 2 64 1 65
19 4 49 3 50 2 51 1 52 0 53 0 54 0 55 0 56
20 0 37 0 38 0 39 0 40 0 41 0 42 0 43 0 44
21 259 316 258 317 257 318 256 319 255 320 254 321 253 322 252 323
22 0 424 0 425 0 426 0 427 0 428 0 429 0 430 0 431
\(i\) \({S}_{i}+48\) \({S}_{i}+50\) \({S}_{i}+52\) \({S}_{i}+54\) \({S}_{i}+56\) \({S}_{i}+58\) \({S}_{i}+60\) \({S}_{i}+62\)
\({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\)
1 0 432 0 433 0 434 0 435 0 436 0 437 0 438 0 439
2 38 92 37 93 36 94 35 95 34 96 33 97 32 98 31 99
3 157 229 156 230 155 231 154 232 153 233 152 234 151 235 150 236
4 282 348 281 349 280 350 279 351 278 352 277 353 276 354 275 355
5 190 241 189 242 188 243 187 244 186 245 185 246 184 247 183 248
6 27 85 26 86 25 87 24 88 23 89 22 90 21 91 20 92
7 78 153 77 154 76 155 75 156 74 157 73 158 72 159 71 160
8 151 210 150 211 149 212 148 213 147 214 146 215 145 216 144 217
9 226 287 225 288 224 289 223 290 222 291 221 292 220 293 219 294
10 0 47 0 48 0 49 0 50 0 51 0 52 0 53 0 54
11 0 73 0 74 0 75 0 76 0 77 0 78 0 79 0 80
12 55 114 54 115 53 116 52 117 51 118 50 119 49 120 48 121
13 54 120 53 121 52 122 51 123 50 124 49 125 48 126 47 127
14 116 178 115 179 114 180 113 181 112 182 111 183 110 184 109 185
15 330 410 329 411 328 412 327 413 326 414 325 415 324 416 323 417
16 18 87 17 88 16 89 15 90 14 91 13 92 12 93 11 94
17 0 37 0 38 0 39 0 40 0 41 0 42 0 43 0 44
18 0 66 0 67 0 68 0 69 0 70 0 71 0 72 0 73
19 0 57 0 58 0 59 0 60 0 61 0 62 0 63 0 64
20 0 45 0 46 0 47 0 48 0 49 0 50 0 51 0 52
21 251 324 250 325 249 326 248 327 247 328 246 329 245 330 244 331
22 0 432 0 433 0 434 0 435 0 436 0 437 0 438 0 439
\(i\) \({S}_{i}+64\) \({S}_{i}+66\) \({S}_{i}+68\) \({S}_{i}+70\) \({S}_{i}+72\) \({S}_{i}+74\) \({S}_{i}+76\) \({S}_{i}+78\)
\({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\)
1 0 440 0 441 0 442 0 443 0 444 0 445 0 446 0 447
2 30 100 29 101 28 102 27 103 26 104 25 105 24 106 23 107
3 149 237 148 238 147 239 146 240 145 241 144 242 143 243 142 244
4 274 356 273 357 272 358 271 359 270 360 269 361 268 362 267 363
5 182 249 181 250 180 251 179 252 178 253 177 254 176 255 175 256
6 19 93 18 94 17 95 16 96 15 97 14 98 13 99 12 100
7 70 161 69 162 68 163 67 164 66 165 65 166 64 167 63 168
8 143 218 142 219 141 220 140 221 139 222 138 223 137 224 136 225
9 218 295 217 296 216 297 215 298 214 299 213 300 212 301 211 302
10 0 55 0 56 0 57 0 58 0 59 0 60 0 61 0 62
11 0 81 0 82 0 83 0 84 0 85 0 86 0 87 0 88
12 47 122 46 123 45 124 44 125 43 126 42 127 41 128 40 129
13 46 128 45 129 44 130 43 131 42 132 41 133 40 134 39 135
14 108 186 107 187 106 188 105 189 104 190 103 191 102 192 101 193
15 322 418 321 419 320 420 319 421 318 422 317 423 316 424 315 425
16 10 95 9 96 8 97 7 98 6 99 5 100 4 101 3 102
17 0 45 0 46 0 47 0 48 0 49 0 50 0 51 0 52
18 0 74 0 75 0 76 0 77 0 78 0 79 0 80 0 81
19 0 65 0 66 0 67 0 68 0 69 0 70 0 71 0 72
20 0 53 0 54 0 55 0 56 0 57 0 58 0 59 0 60
21 243 332 242 333 241 334 240 335 239 336 238 337 237 338 236 339
22 0 440 0 441 0 442 0 443 0 444 0 445 0 446 0 447
\(i\) \({S}_{i}+80\) \({S}_{i}+82\) \({S}_{i}+84\) \({S}_{i}+86\) \({S}_{i}+88\) \({S}_{i}+90\) \({S}_{i}+92\) \({S}_{i}+94\)
\({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\)
1 0 448 0 449 0 450 0 451 0 452 0 453 0 454 0 455
2 22 108 21 109 20 110 19 111 18 112 17 113 16 114 15 115
3 141 245 140 246 139 247 138 248 137 249 136 250 135 251 134 252
4 266 364 265 365 264 366 263 367 262 368 261 369 260 370 259 371
5 174 257 173 258 172 259 171 260 170 261 169 262 168 263 167 264
6 11 101 10 102 9 103 8 104 7 105 6 106 5 107 4 108
7 62 169 61 170 60 171 59 172 58 173 57 174 56 175 55 176
8 135 226 134 227 133 228 132 229 131 230 130 231 129 232 128 233
9 210 303 209 304 208 305 207 306 206 307 205 308 204 309 203 310
10 0 63 0 64 0 65 0 66 0 67 0 68 0 69 0 70
11 0 89 0 90 0 91 0 92 0 93 0 94 0 95 0 96
12 39 130 38 131 37 132 36 133 35 134 34 135 33 136 32 137
13 38 136 37 137 36 138 35 139 34 140 33 141 32 142 31 143
14 100 194 99 195 98 196 97 197 96 198 95 199 94 200 93 201
15 314 426 313 427 312 428 311 429 310 430 309 431 308 432 307 433
16 2 103 1 104 0 105 0 106 0 107 0 108 0 109 0 110
17 0 53 0 54 0 55 0 56 0 57 0 58 0 59 0 60
18 0 82 0 83 0 84 0 85 0 86 0 87 0 88 0 89
19 0 73 0 74 0 75 0 76 0 77 0 78 0 79 0 80
20 0 61 0 62 0 63 0 64 0 65 0 66 0 67 0 68
21 235 340 234 341 233 342 232 343 231 344 230 345 229 346 228 347
22 0 448 0 449 0 450 0 451 0 452 0 453 0 454 0 455
\(i\) \({S}_{i}+96\) \({S}_{i}+98\) \({S}_{i}+100\)
\({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\) \({a}_{i}^{*}\) \({b}_{i}^{*}\)
1 0 456 0 457 0 458
2 14 116 13 117 12 118
3 133 253 132 254 131 255
4 258 372 257 373 256 374
5 166 265 165 266 164 267
6 3 109 2 110 1 111
7 54 177 53 178 52 179
8 127 234 126 235 125 236
9 202 311 201 312 200 313
10 0 71 0 72 0 73
11 0 97 0 98 0 99
12 31 138 30 139 29 140
13 30 144 29 145 28 146
14 92 202 91 203 90 204
15 306 434 305 435 304 436
16 0 111 0 112 0 113
17 0 61 0 62 0 63
18 0 90 0 91 0 92
19 0 81 0 82 0 83
20 0 69 0 70 0 71
21 227 348 226 349 225 350
22 0 456 0 457 0 458

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Budak, G., Chen, X. Evaluation of the size of time windows for the travelling salesman problem in delivery operations. Complex Intell. Syst. (2020). https://doi.org/10.1007/s40747-020-00167-y

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Keywords

  • Operational research
  • Package delivery
  • Time windows
  • Travelling salesman problem