Determination of optimal depot location for a capacitated vehicle routing problem (CVRP) based on gross vehicle weight

Figures & data

Figure 1. Travel economy of a truck for its various gross weights.

Source: Reproduced (Rahman et al., 2022).

Figure 2. (a) The geographical locations of wholesalers (b) within the western region of Bangladesh.

Table 1. Distance matrix.

	FAR	MAG	RAJ	KUS	CHO	JHE	JES	NAR
FAR	0	31	30	95	117	80	96	92
MAG	31	0	32	73	65	29	46	45
RAJ	30	32	0	65	120	83	100	118
KUS	95	73	65	0	50	46	93	117
CHO	117	65	120	50	0	36	75	108
JHE	80	29	83	46	36	0	45	79
JES	96	46	100	93	75	45	0	33
NAR	92	45	118	117	108	79	33	0

Table 2. Weekly demands of the wholesalers.

Wholesalers	FAR	MAG	RAJ	KUS	CHO	JHE	JES	NAR
Weekly demand (tons)	1.5	0.5	1.5	3.0	1.5	0.5	1.0	0.5

Figure 3. The research framework.

Table 3. List of parameters and decision variables used in mathematical modelling.

Parameters
N	Set of wholesalers
$F_{\mathrm{ij}}$	Consumption of fuel from a wholesaler $\left(i\in N\right)$ to another $\left(j\in N\right)$
Decision variables
$x_{\mathrm{ij}}\in \left\{0,1\right\}$	If the vehicle travels from $i\in N$ to $j\in N$ then 1, otherwise 0
$u_i\in Z+$	A value that connects $x_{\mathrm{ij}}$ to the next node

Table 4. Description of the variables. 

Variables	Description
	$\begin{array}{l}\text{Fuel consumption in delivery route}\\ \text{Fuel consumption in return route}\\ \text{Total fuel consumption}\end{array}\}L$
$f_{\mathrm{ld}}$
$f_{\mathrm{lr}}$
$C_{\mathrm{tf}}$
	$\begin{array}{l}\text{Travel distance}\\ \text{Return distance from last wholesaler to DL}\\ \text{Total travelling distance}\end{array}\}\mathrm{km}$
$d$
$R_d$
D
x	Gross weight of a vehicle, ton
$C_l$	Cost of fuel per litre, tk./L
$y$	Travel economy, km/L
	$\begin{array}{l}\text{Transportation costs in delivery route}\\ \text{Transportation costs in return route}\\ \text{Total transportation cost}\end{array}\}\mathrm{Tk}$
$T{C}_D$
$T{C}_R$
$\mathrm{TTC}$
$\mathrm{TT}{C}_m$	Minimum transportation cost
$\mathrm{TT}{C}_P$	$\text{Cost performance }\left(\text{\%}\right)$

Table 5. Fuel consumption matrix for the DL at FAR.

	FAR	MAG	RAJ	KUS	CHO	JHE	JES	NAR
FAR	0	10.84	10.00	27.81	27.53	17.14	19.97	18.09
MAG	10.84	0	10.67	21.37	15.29	6.21	9.57	8.85
RAJ	10.00	10.67	0	19.03	28.24	17.78	20.81	23.21
KUS	27.81	21.37	19.03	0	11.76	9.85	19.35	23.01
CHO	27.53	15.29	28.24	11.76	0	7.71	15.60	21.24
JHE	17.14	6.21	17.78	9.85	7.71	0	9.36	15.54
JES	19.97	9.57	20.81	19.35	15.60	9.36	0	6.49
NAR	18.09	8.85	23.21	23.01	21.24	15.54	6.49	0

Table 6. Fuel consumption, travel route, and path distance for each depot.

DL	Travel route	Path distance (km)	Fuel consumption (L)
FAR	1-3-4-5-6-7-8-2-1	335	79.168
MAG	1-8-7-6-5-4-3-2-1	335	79.168
RAJ	1-4-5-6-7-8-2-3-1	335	79.310
KUS	1-5-6-7-8-3-4-2-1	335	72.558
CHO	1-7-8-4-5-3-2-6-1	335	79.662
JHE	1-2-3-4-6-5-8-7-1	335	85.450
JES	1-8-6-7-5-4-3-2-1	335	80.703
NAR	1-2-3-4-5-6-8-7-1	335	91.782

Figure 4. Transportation routes in (a) nearest neighbour heuristic (b) sweeping heuristic and (c) Clarke & Wright's heuristic.

Table 7. Heuristics’ algorithms.

Nearest neighbour heuristic	Sweeping heuristic	Clarke & Wright’s heuristic
Input set of customers $S_i=\left\{1,2,\dots \dots \dots ,n\right\}$Output routes $i=\left\{1,2,\dots ,n\right\}$ for $S_1$ = 1, 2, … , n; and $d=0$ do $D_{\mathrm{dj}}=min\left\{D_{\mathrm{dj}}\right\}j\in S_1$ Now, $d=j$ and unload $q_j$ at location j $S_1=S_1-j$ and $Q=Q-q_j$  while $\left(Q\ne 0\right)$ do repeat 2   if $\left(Q=0\right)$ then  back to DL and get route $i=1$   end if  end while  while $\left(S_1\ne 0\right)$ do repeat 2–4, and get routes $i=2,\dots ,n$   if $\left(S_1=0\right)$ then back to DL    end if   end while  end for return $S_2,S_3,\dots \dots .,S_n$	Input set of customers $S_i=\left\{1,2,\dots \dots \dots ,n\right\}$ Output routes $i=\left\{1,2,\dots ,n\right\}$ for $S_1$ = 1, 2, … , n; and $d=0$ do Turns a ray counterclockwise from the DL. Connect d with j that is crossed by the ray,  $j\in S_1$ Now, $d=j$ and unload $q_j$ at location j $S_1=S_1-j$ and $Q=Q-q_j$  while $\left(Q\ne 0\right)$ do repeat 3   if $\left(Q=0\right)$ then back to DL and get route $i=1$   end if end while  while $\left(S_1\ne 0\right)$ do repeat 3–5, and get routes $i=2,\dots ,n$   if $\left(S_1=0\right)$ then back to DL   end if  end while end forreturn $S_2,S_3,\dots \dots .,S_n$	Input set of customers $S_i=\left\{1,2,\dots \dots \dots ,n\right\}$ Output routes $i=\left\{1,2,\dots ,n\right\}$ for $S_1$ = 1, 2, … , n; and $d=0$ do $C_{\mathrm{dj}}=min\left\{C_{\mathrm{dj}}\right\}j\in S_1$ Now, $d=j$ and unload $q_j$ at location j $S_1=S_1-j$ and $Q=Q-q_j$  while $\left(Q\ne 0\right)$ do  repeat 2   if $\left(Q=0\right)$ then   back to DL and get route $i=1$   end if  end while  while $\left(S_1\ne 0\right)$ do  repeat 2–4, and get routes  $i=2,\dots ,n$   if $\left(S_1=0\right)$ then   back to DL   end if  end while  end forreturn $S_2,S_3,\dots \dots .,S_n$
Where $D$ is distance, $C$ is cost, $d$ is DL location, $Q$ is vehicle capacity, $q_j$ is the demand of a customer

Table 8. Routes, path distances, and fuel consumptions for the heuristics.

Heuristics	Route	Path distance (km)	Fuel consumption (L)
NNH	1-2-3-4-5-6-7-8-1	384	93.988
SH	1-3-4-5-6-2-7-8-1	381	88.427
CWH	1-2-8-7-6-5-4-3-1	335	79.815

Figure 5. Comparison of GVWs between (a) KUS and FAR, (b) KUS and MAG, (c) KUS and RAJ, (d) KUS and CHO, (e) KUS and JHE, (f) KUS and JES, and (g) KUS and NAR

Figure 6. Fuel consumption comparison for each DL and heuristic method.

Table 9. Various quantities of loads (tons) are to be delivered to the wholesalers for different instances.

Instances	KUS	RAJ	MAG	FAR	CHO	JHE	JES	NAR
1	0.5	1	1.5	1	2.5	1.5	1	1
2	1	1	1.5	1	2	1.5	1	1
3	1.5	1	1.5	1	1.5	1.5	1	1
4	2	1	1.5	1	1.5	1	1	1
5	2.5	1	1.5	1	1.5	1	1	0.5
6	3	1.5	0.5	1.5	1.5	0.5	1	0.5
7	3.5	1	0.5	1.5	1	1	1	0.5
8	4	1	0.5	1	1	1	1	0.5

Table 10. Fuel consumption for each depot in various instances.

Inst.	DL	$f_{\mathrm{ld}}$	$f_{\mathrm{lr}}$	$C_{\mathrm{tf}}$	Inst.	DL	$f_{\mathrm{ld}}$	$f_{\mathrm{lr}}$	$C_{\mathrm{tf}}$
1	KUS	66.586	12.452	79.039	2	KUS	75.651	9.578	85.229
	FAR	76.556	5.938	82.495		FAR	76.027	5.938	81.966
	MAG	76.557	5.938	82.496		MAG	76.557	5.938	82.496
	RAJ	67.405	12.452	79.857		RAJ	76.346	5.747	82.093
	CHO	76.557	6.896	83.454		CHO	77.297	6.896	84.194
	JHE	70.988	6.896	77.885		JHE	74.505	8.620	83.126
	JES	69.736	8.620	78.357		JES	80.105	6.321	86.427
	NAR	77.479	8.620	86.100		NAR	77.899	8.620	86.520
3	KUS	64.103	12.452	76.556	4	KUS	62.732	12.452	75.185
	FAR	75.536	5.938	81.475		FAR	74.811	5.938	80.750
	MAG	75.536	5.938	81.475		MAG	74.811	5.938	80.750
	RAJ	75.855	5.747	81.602		RAJ	75.130	5.747	80.877
	CHO	77.877	6.896	84.774		CHO	76.206	6.896	83.103
	JHE	75.130	8.620	83.751		JHE	76.450	8.620	85.071
	JES	80.630	6.321	86.952		JES	81.734	6.321	88.056
	NAR	78.365	8.620	86.986		NAR	79.303	8.620	87.924
5	KUS	60.745	12.452	73.197	6	KUS	60.111	12.447	72.558
	FAR	73.423	5.938	79.362		FAR	73.232	5.936	79.168
	MAG	73.423	5.938	79.362		MAG	73.232	5.936	79.168
	RAJ	73.742	5.747	79.489		RAJ	73.565	5.745	79.310
	CHO	73.758	6.896	80.655		CHO	72.768	6.894	79.662
	JHE	74.953	8.620	83.574		JHE	76.833	8.617	85.450
	JES	80.5539	6.321	86.875		JES	72.086	8.617	80.703
	NAR	78.858	6.321	85.180		NAR	83.165	8.617	91.782
7	KUS	60.346	12.452	72.799	8	KUS	59.791	12.452	72.243
	FAR	74.276	5.938	80.215		FAR	75.642	5.938	81.581
	MAG	74.276	5.938	80.215		MAG	75.641	5.938	81.580
	RAJ	75.112	5.747	80.859		RAJ	75.991	5.747	81.738
	CHO	73.555	6.896	80.452		CHO	72.431	6.896	79.328
	JHE	71.656	6.896	78.552		JHE	74.545	8.620	83.166
	JES	70.985	8.620	79.606		JES	70.122	8.620	78.743
	NAR	79.006	6.321	85.328		NAR	77.915	6.321	84.237

Table 11. Optimal DL for each instance.

Instances	1	2	3	4	5	6	7	8
Optimal DL	JHE	FAR	KUS	KUS	KUS	KUS	KUS	KUS

Table A1. The distance matrix for the DL at KUS.

	KUS	RAJ	MAG	FAR	CHO	JHE	JES	NAR
KUS	0	50	120	65	117	36	75	108
RAJ	50	0	65	73	95	46	93	117
MAG	120	65	0	32	30	83	100	118
FAR	65	73	32	0	31	29	46	45
CHO	117	95	30	31	0	80	96	92
JHE	36	46	83	29	80	0	45	79
JES	75	93	100	46	96	45	0	33
NAR	108	117	118	45	92	79	33	0

Table A2. The fuel consumption matrix for the DL at KUS.

	KUS	RAJ	MAG	FAR	CHO	JHE	JES	NAR
KUS	0	19.84	19.76	24.79	11.76	9.85	19.35	23.01
RAJ	19.84	0	8.66	7.82	28.24	17.78	20.81	23.21
MAG	19.76	8.66	0	8.08	15.29	6.21	9.57	8.85
FAR	24.79	7.82	8.08	0	27.53	17.14	19.97	18.09
CHO	11.76	28.24	15.29	27.53	0	7.71	15.60	21.24
JHE	9.85	17.78	6.21	17.14	7.71	0	9.36	15.54
JES	19.35	20.81	9.57	19.97	15.60	9.36	0	6.49
NAR	23.01	23.21	8.85	18.09	21.24	15.54	6.49	0

Rahman, M. H., Fashiar Rahman, M., & Tseng, T. L. (2022). Estimation of fuel consumption and selection of the most carbon-efficient route for cold-chain logistics. International Journal of Systems Science: Operations & Logistics, 10(1), 1–17.

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Data availability

The authors confirm that most of the data is available within the article, and the remaining data used in the OPL CPLEX 22.1.1 will be available on request.