A Novel Travel Time Estimation Model for Modeling a Green Time-Dependent Vehicle Routing Problem in Food Supply Chain
Abstract
:1. Introduction
2. Literature Review
Algorithm 1: Travel Time Measurement |
$1.sett-to-{t}_{o},$ $setd-to-{d}_{ij},$ $set-{t}^{\prime}-to-t+\left(\frac{d}{{v}_{c}{T}_{k}}\right).$ $2.while\left({t}^{\prime}{\overline{t}}_{k}\right)do$ $2.1.d\leftarrow d-{v}_{c}{T}_{k}\left({\overline{t}}_{k}-t\right),$ $2.2.t\leftarrow {\overline{t}}_{k},$ $2.3.{t}^{\prime}\leftarrow t+\left(\frac{d}{{v}_{c}{T}_{k+1}}\right),$ $2.4.k\leftarrow k+1.$ $3.return\left({t}^{\prime}-{t}_{o}\right).$ |
3. Methodology
Algorithm 2: Pseudo code of calculating travel time between two consecutive nodes (i,j). |
Input: i,j, t, ${t}_{i}^{k}$, s,$A{V}_{ijs}^{t}$,${d}_{ijs}$,${\tau}_{s}$, ${U}_{t}$; Output: ${T}_{ij}$, 1: set $0\leftarrow {\tau}_{o}$, ${t}_{o}\leftarrow {t}_{i}^{k}$ 2: for $s=1$ to $thenumberof\mathrm{se}gmentsbetweennodeiandj$ 3: set ${t}^{\u2033}\leftarrow {t}_{o}+{\tau}_{s-1}$ 4: find t in which departure time of the vehicle (${t}_{o}$) has occurred (${U}_{t},{U}_{t+1}$) 5: if (${t}^{\u2033}<{U}_{t+1}$) then 6: set ${t}_{o}\leftarrow {t}^{\u2033}$ 7: set $d\leftarrow {d}_{ijs}$ 8: set ${t}^{\prime}\leftarrow {t}_{o}+\frac{d}{A{V}_{ijs}^{t}}$ 9: if (${t}^{\prime}\le {U}_{t+1}$) then 10: set ${\mathit{T}}_{\mathit{i}\mathit{j}\mathit{s}}\leftarrow \left({\mathit{t}}^{\prime}-{\mathit{t}}^{\u2033}\right)$ 11: else 12: while (${t}^{\prime}>{U}_{t+1}$) do 13: set $\mathit{d}\leftarrow \mathit{d}-\mathit{A}{\mathit{V}}_{\mathit{i}\mathit{j}\mathit{s}}^{\mathit{t}}\left({\mathit{U}}_{\mathit{t}+\mathit{1}}-{\mathit{t}}_{\mathit{o}}\right)$ 14: set ${t}_{o}\leftarrow {U}_{t+1}$ 15: set ${\mathit{t}}^{\prime}\leftarrow {\mathit{t}}_{\mathit{o}}+\frac{d}{\mathit{A}{\mathit{V}}_{\mathit{i}\mathit{j}\mathit{s}}^{\mathit{t}+\mathit{1}}}$ 16: set $t\leftarrow t+1$ 17: end while 18: set ${\mathit{T}}_{\mathit{i}\mathit{j}\mathit{s}}\leftarrow \left({\mathit{t}}^{\prime}-{\mathit{t}}^{\u2033}\right)$ 19: end ifv20: else 21: set $\mathit{t}\leftarrow \mathit{t}+\mathit{1}$ 22: set ${\mathit{t}}_{\mathit{o}}\leftarrow {\mathit{t}}^{\u2033}$ 23: set $\mathit{d}\leftarrow {\mathit{d}}_{\mathit{i}\mathit{j}\mathit{s}}$ 24: set ${\mathit{t}}^{\prime}\leftarrow {\mathit{t}}_{\mathit{o}}+\frac{\mathit{d}}{\mathit{A}{\mathit{V}}_{\mathit{i}\mathit{j}\mathit{s}}^{\mathit{t}}}$ 25: if (${t}^{\prime}\le {U}_{t+1}$) then 26: set ${\mathit{T}}_{\mathit{i}\mathit{j}\mathit{s}}\leftarrow \left({\mathit{t}}^{\prime}-{\mathit{t}}^{\u2033}\right)$ 27: else 28: while (${t}^{\prime}>{U}_{t+1}$) do 29: set $\mathit{d}\leftarrow \mathit{d}-\mathit{A}{\mathit{V}}_{\mathit{i}\mathit{j}\mathit{s}}^{\mathit{t}}\left({\mathit{U}}_{\mathit{t}+\mathit{1}}-{\mathit{t}}_{\mathit{o}}\right)$ 30: set ${t}_{o}\leftarrow {U}_{t+1}$ 31: set ${\mathit{t}}^{\prime}\leftarrow {\mathit{t}}_{\mathit{o}}+\frac{\mathit{d}}{\mathit{A}{\mathit{V}}_{\mathit{i}\mathit{j}\mathit{s}}^{\mathit{t}+\mathit{1}}}$ 32: set $t\leftarrow t+1$ 33: end while 34: set ${\mathit{T}}_{\mathit{i}\mathit{j}\mathit{s}}\leftarrow \left({\mathit{t}}^{\prime}-{\mathit{t}}^{\u2033}\right)$ 35: end if 36: end if 37: save ${\mathit{T}}_{\mathit{i}\mathit{j}\mathit{s}}$ 38: set ${\mathit{t}}_{\mathit{o}}\leftarrow {\mathit{t}}^{\prime}$ 39: end for 40: set ${\mathit{T}}_{\mathit{i}\mathit{j}}\leftarrow {\displaystyle \sum}_{\mathit{S}}{\mathit{T}}_{\mathit{i}\mathit{j}\mathit{s}}$ |
4. Case Study and Results
- Congestion highly depends on the road type. The congestion pattern was not similar in different road types. For instance, Figure 5 reveals that the congestion pattern of the route between customer 6 and the depot varied from one section (Esteghlal Boulevard) to another section (Hengam Avenue). On Esteghlal Boulevard, the free-flow traffic mode had the largest share in all periods; however, this amount fluctuated between 70% and 92%. High congestion in both sections was not anticipated during the given traffic period, where in the Hengam Avenue section, free-flow traffic mode had the largest share until 9:00 and then low congestion replaced it until the end of the traffic period (i.e., 15:00).
- Congestion patterns in the same road types were different. As an illustration (Figure 6), on Zeinodin Expressway, the pattern was uniform and only a free-flow traffic mode existed during the given traffic period. Nevertheless, both the free-flow mode and low congestion existed on Baqeri Expressway.
- Congestion highly depended on the time. Each section of the route in different traffic modes illustrated different behaviour during the whole period, as follows.
- ○
- Uniform behaviour (i.e., free-flow traffic mode on Zeinodin Expressway)
- ○
- Sinusoidal behaviour (i.e., free-flow traffic mode on Baqeri Expressway)
- ○
- Cosine behaviour (i.e., low congestion on Baqeri Expressway)
- ○
- Ascending behaviour (i.e., low congestion on Hengam Avenue)
- ○
- Descending behaviour (i.e., free-flow traffic mode on Hengam Avenue)
- ○
- Oscillating behaviour (i.e., free-flow traffic mode and low congestion on Esteghlal Boulevard)
5. Discussion
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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References | Approach | Congestion Period | Case Study | Speed Profile | Intermediate Node | Multiple Traffic Modes | Route Dependency | |||||||||
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JM | M and A | M | JA | S | I | D | V | $\Delta $ | R | C | ||||||
[5] | Stochastic | * | ||||||||||||||
[35] | continuous | * | * | * | ||||||||||||
[36] | continuous | * | * | * | ||||||||||||
[20] | continuous | * | * | * | * | |||||||||||
[8] | Discrete | * | * | |||||||||||||
[26] | continuous | * | * | * | * | |||||||||||
[37] | continuous | * | * | * | ||||||||||||
[25] | continuous | * | * | |||||||||||||
[38] | continuous | * | * | |||||||||||||
[16] | Stochastic | * | * | |||||||||||||
[29] | continuous | * | * | * | * | |||||||||||
[39] | continuous | * | * | * | ||||||||||||
[40] | continuous | * | * | |||||||||||||
[17] | Stochastic | * | * | |||||||||||||
[22] | continuous | * | * | * | ||||||||||||
[6] | Stochastic | * | * | |||||||||||||
[23] | continuous | * | * | * | * | |||||||||||
[41] | continuous | * | * | * | ||||||||||||
[10] | continuous | * | * | |||||||||||||
[13] | continuous | * | * | * | * | * | ||||||||||
[11] | continuous | * | * | |||||||||||||
[27] | continuous | * | * | |||||||||||||
[42] | continuous | * | * | * | ||||||||||||
[43] | continuous | * | * | |||||||||||||
This paper | continuous | * | * | * | * | * | * | * | * | * | * | * |
Notation | Definition |
---|---|
N | Nodes related to warehouse and customers (a warehouse is indicated by o) |
i, j | Nodes |
f | Four traffic modes include (1) free-flow mode (green colour), (2) low congestion (orange colour), (3) medium congestion (red colour), and (4) high congestion (crimson colour) based on Google Maps. |
s | Route segment |
t | Traffic period. Here, each day is divided into different time intervals. Each time interval represents one traffic period. For example, third time interval is the third traffic period. |
${t}_{i}^{k}$ | The departure time of vehicle k from ith node |
${v}_{ijsf}$ | Speed (km/h) at segment s of the route from node i to j in traffic mode f |
$A{V}_{ijs}^{t}$ | The average speed at segment s of the route from node i to j in traffic mode f |
${T}_{ijs}$ | Travel time of segment s of the route from node i to j |
S | The set of all urban roads (alley, auxiliary street, main street, boulevard, and expressway) |
${d}_{ijs}$ | Distance of segment s of the route from node i to j (meter) |
${\tau}_{s}$ | Transition time from segment s to s+1 on the route from node i to j (equal to intermediate nodes on the route from node i to j) (seconds) |
$r$ | Number of traffic periods |
${U}_{t}$ | $\mathrm{Bottom}\mathrm{of}\mathrm{the}\mathrm{range}\mathrm{in}\mathrm{traffic}\mathrm{period}t(\mathrm{the}\mathrm{range}\mathrm{for}\mathrm{this}\mathrm{period}\mathrm{is}\mathrm{equal}\mathrm{to}({U}_{t},{U}_{t+1}]$) |
${l}_{ijsf}^{t}$ | Percentage of the traversed distance in segment s of the route from node i to j in traffic mode f and traffic period t |
No. | Road Type (S) | Average Speed (km/h) in Congestion Modes | |||
---|---|---|---|---|---|
Congestion | |||||
Green (Free-Flow) | Orange (Low) | Red (Medium) | Crimson (High) | ||
1 | Expressway | 50–60 | 20 | 10 | 5 |
2 | Boulevard | 40–50 | 20 | 10 | 5 |
3 | Main Street | 30–40 | 20 | 10 | 5 |
4 | Auxiliary Street | 20–30 | 20 | 10 | 5 |
5 | Alley | 15 | …… | …… | …… |
No. | Route | Departure Time | Proposed Model Estimation (min) | Without Intermediate Node, Multiple Traffic Modes, and Route Dependency | Google Estimation (min) | |||
---|---|---|---|---|---|---|---|---|
Hour | Minute | |||||||
Alinaghian & Naderipour [23] | Naderipour & Alinaghian [22] | |||||||
1 | 1-2 | 7 | 33 | 3.4706 | 1.95 | 1.733333 | 4 | |
2 | 0-2 | 13 | 6 | 10.1396 | 7.2 | 4.430769 | 10 | |
3 | 5-1 | 13 | 47 | 5.2694 | 4.95 | 3.046154 | 5 | |
4 | 1-12 | 13 | 51 | 9.1104 | 8.4 | 5.169231 | 9 | |
5 | 9-2 | 11 | 44 | 6.4734 | 2.4 | 2.181818 | 7 | |
6 | 3-4 | 7 | 43 | 8.1017 | 7.05 | 6.266667 | 7 | |
7 | 3-15 | 9 | 5 | 7.8736 | 6 | 5.454545 | 8 | |
8 | 7-4 | 11 | 6 | 3.1863 | 1.92 | 1.745455 | 4 | |
9 | 4-9 | 14 | 10 | 10.1989 | 7.05 | 4.338462 | 10 | |
10 | 10-5 | 14 | 14 | 13.3933 | 10.35 | 6.369231 | 13 | |
11 | 9-6 | 13 | 6 | 11.211 | 8.85 | 5.446154 | 11 | |
12 | 6-11 | 13 | 47 | 8.1976 | 4.95 | 3.046154 | 7 | |
13 | 8-7 | 7 | 57 | 8.0336 | 7.8 | 6.933333 | 7 | |
14 | 11-7 | 13 | 6 | 6.277 | 6.6 | 4.061538 | 6 | |
15 | 12-8 | 13 | 47 | 2.8682 | 1.8 | 1.107692 | 4 | |
16 | 15-8 | 13 | 51 | 3.7801 | 2.55 | 1.569231 | 4 | |
17 | 12-9 | 11 | 44 | 10.0803 | 4.2 | 3.818182 | 12 | |
18 | 11-10 | 7 | 43 | 5.1244 | 3.3 | 2.933333 | 6 | |
19 | 16-10 | 9 | 5 | 1.454 | 1.08 | 0.981818 | 2 | |
20 | 14-11 | 11 | 6 | 2.9711 | 1.56 | 1.418182 | 3 | |
21 | 16-11 | 14 | 10 | 2.7063 | 1.8 | 1.107692 | 3 | |
22 | 11-16 | 14 | 14 | 3.1024 | 1.95 | 1.2 | 5 | |
23 | 13-12 | 8 | 10 | 11.2049 | 9.9 | 8.8 | 12 | |
24 | 16-12 | 14 | 16 | 9.8184 | 8.7 | 5.353846 | 10 | |
25 | 14-13 | 14 | 10 | 3.2247 | 2.1 | 1.292308 | 5 | |
26 | 13-14 | 10 | 38 | 2.6125 | 1.14 | 1.036364 | 3 | |
27 | 13-16 | 13 | 0 | 2.2063 | 1.2 | 0.738462 | 3 | |
28 | 15-14 | 8 | 3 | 4.4721 | 3.15 | 2.8 | 6 | |
29 | 16-14 | 10 | 9 | 0.5761 | 0.264 | 0.24 | 1 | |
30 | 16-15 | 13 | 52 | 9.7121 | 8.55 | 5.261538 | 8 |
Paired Differences | T | Df | Sig. (2-Tailed) | ||||||
---|---|---|---|---|---|---|---|---|---|
Mean | Std. Deviation | Std. Error Mean | 95% Confidence Interval of the Difference | ||||||
Lower | Upper | ||||||||
Pair 1 | Google—Proposed | 0.271657 | 0.889936 | 0.162479 | −0.060651 | 0.603964 | 1.672 | 29 | 0.105 |
Paired Differences | T | Df | Sig. (2-Tailed) | ||||||
---|---|---|---|---|---|---|---|---|---|
Mean | Std. Deviation | Std. Error Mean | 95% Confidence Interval of the Difference | ||||||
Lower | Upper | ||||||||
Pair 1 | Google—Alinaghian | 1.876200 | 1.668691 | 0.304660 | 1.253100 | 2.499300 | 6.158 | 29 | 0.000 |
Paired Differences | t | df | Sig. (2-Tailed) | ||||||
---|---|---|---|---|---|---|---|---|---|
Mean | Std. Deviation | Std. Error Mean | 95% Confidence Interval of the Difference | ||||||
Lower | Upper | ||||||||
Pair 1 | Google—Naderipour | 3.170617 | 1.853967 | 0.338486 | 2.478334 | 3.862900 | 9.367 | 29 | 0.000 |
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Asgharizadeh, E.; Jooybar, S.; Mahdiraji, H.A.; Garza-Reyes, J.A. A Novel Travel Time Estimation Model for Modeling a Green Time-Dependent Vehicle Routing Problem in Food Supply Chain. Sustainability 2022, 14, 8633. https://doi.org/10.3390/su14148633
Asgharizadeh E, Jooybar S, Mahdiraji HA, Garza-Reyes JA. A Novel Travel Time Estimation Model for Modeling a Green Time-Dependent Vehicle Routing Problem in Food Supply Chain. Sustainability. 2022; 14(14):8633. https://doi.org/10.3390/su14148633
Chicago/Turabian StyleAsgharizadeh, Ezzatollah, Sobhan Jooybar, Hannan Amoozad Mahdiraji, and Jose Arturo Garza-Reyes. 2022. "A Novel Travel Time Estimation Model for Modeling a Green Time-Dependent Vehicle Routing Problem in Food Supply Chain" Sustainability 14, no. 14: 8633. https://doi.org/10.3390/su14148633