A multi-path routing-inventory problem for a closed-loop supply chain considering the heterogeneous fleet of vehicles

Figures & data

Figure 1. A typical schema for the problem considered in this study.

Table 1. Comparison between the features of several recent studies and the current study.

Author(s)	Type of SC	Road roughness degree	Multi- path	Holding cost	Limited number of vehicles	Capacity of vehicles	Heterogeneous fleet of vehicles	Multi- period
Tosarkani and Amin (2018)	CLSC	-	-	†	-	-	-	†
Kaya and Urek (2016)	CLSC	-	-	†	-	-	-	-
Bazan, Jaber, and Zanoni (2017)	CLSC	-	-	†	-	-	-	-
Kannan et al. (2012)	Reverse SC	-	-	-	-	-	-	-
Shankar, Bhattacharyya, and Choudhary (2018)	CLSC	-	-	†	-	-	-	†
Üster and Hwang (2017)	CLSC	-	-	-	-	-	-	-
Fahimnia et al. (2013a)	Forward SC	†	-	-	-	-	-	†
Fahimnia et al. (2015)	Forward SC	-	-	†	-	-	†	†
Rahmani and Mahoodian (2017)	Forward SC	-	-	-	-	-	†	-
Talaei et al. (2016)	CLSC	-	-	-	-	†	-	-
Fahimnia et al. (2013b)	CLSC	-	-	†	-	-	†	†
Tao et al. (2015)	CLSC	-	-	†	-	-	-	†
Zakeri et al. (2015)	CLSC	-	-	†	-	†	-	†
Aalaei and Davoudpour (2017)	Forward SC	-	-	†	-	-	-	-
Soysal et al. (2015)	CLSC	-	-	†	†	†	-	†
Soysal (2016)	CLSC	-	-	†	†	†	-	†
Choudhary et al. (2015)	CLSC	-	-	-	-	†	-	-
Mohammed et al. (2017)	CLSC	-	-	†	-	†	-	†
Keshari et al. (2017)	Forward SC	-	-	†	-	-	†	-
Alhaj, Svetinovic, and Diabat (2016)	Forward SC	-	-	†	-	-	-	-
Rahimi, Baboli, and Rekik (2017)	CLSC	-	-	†	-	†	†	†
Iassinovskaia, Limbourg, and Riane (2017)	CLSC	-	-	†	†	†	-	†
Zhang, Alshraideh, and Diabat (2018)	Reverse SC	-	-	†	†	†	†	†
Archetti, Christiansen, and Grazia Speranza (2018)	Forward SC	-	-	†	†	†	-	†
Current study	CLSC	†	†	†	†	†	†	†

Table 2. Mathematical formulations for the number of decision variables and constraints.

Variables	Binary variables	${\left|\mathcal{I}\right|}^2.\left|\mathcal{V}\right|.\left|\mathcal{P}\right|.\left|\mathcal{T}\right|$
	Positive integer variables	$2.\left|\mathcal{I}\right|.\left|\mathcal{T}\right|\left(\left|\mathcal{I}\right|.\left|\mathcal{V}\right|.\left|\mathcal{P}\right|+1\right)$
	Positive variables	$\left|\mathcal{I}\right|.\left|\mathcal{V}\right|.\left|\mathcal{T}\right|$
	Total number of variables	$\left|\mathcal{I}\right|.\left|\mathcal{T}\right|\left(\left|\mathcal{V}\right|\left[\left|\mathcal{I}\right|.\left|\mathcal{P}\right|+1\right]+2\right)$
Constraints	Equality constraints	$\left|\mathcal{T}\right|(\left|\mathcal{S}\right|.|$E$|+\left|\mathcal{G}\right|\left[\left|\mathcal{R}\right|+\left|\mathcal{D}\right|+\left|\mathcal{M}\right|\right]+$ $2.\left|\mathcal{R}\right|+\left|\mathcal{D}\right|+\left|\mathcal{M}\right|)$
	Inequality constraints	$\left|\mathcal{T}\right|(\left|\mathcal{G}\right|\left[4.\left|\mathcal{M}\right|+2.\left|\mathcal{D}\right|+\left|\mathcal{R}\right|\left(\left|\mathcal{R}\right|.\left|\mathcal{P}\right|+4\right)\right]+$ $\left|\mathcal{S}\right|[4.|$E$\left|\mathcal{S}\right|[4.|$
	Total number of constraints	$\left|\mathcal{T}\right|(\left|\mathcal{G}\right|\left[5.\left|\mathcal{M}\right|+3.\left|\mathcal{D}\right|+\left|\mathcal{R}\right|\left(\left|\mathcal{R}\right|.\left|\mathcal{P}\right|+5\right)\right]+$ $\left|\mathcal{S}\right|[5.|$E$|+1]+4.\left|ℛ\right|+2.\left|ℳ\right|+\left|\mathcal{D}\right|)$

Table 3. Specification of instances for sensitivity analysis on the number of decision variables and constraints.

Specification Instance	$\left|\mathcal{S}\right|$	$\left|\mathcal{M}\right|$	$\left|\mathcal{R}\right|$	$\left|\mathcal{D}\right|$	$|$E$|$	$\left|\mathcal{G}\right|$	$\left|\mathcal{P}\right|$	$\left|\mathcal{T}\right|$
X	1	2	5	1	2	3	1	1
2x	2	4	10	2	4	6	2	2
3x	3	6	15	3	6	9	3	3
4x	4	8	20	4	8	12	4	4
5x	5	10	25	5	10	15	5	5

Figure 2. Comparison between the number of variables and constraints.

Table 4. The results obtained for the instances.

	Specification
Instance	$\left|\mathcal{S}\right|$	$\left|\mathcal{M}\right|$	$\left|\mathcal{R}\right|$	$\left|\mathcal{D}\right|$	$|$E$|$	$\left|\mathcal{G}\right|$	$\left|\mathcal{T}\right|$	No. of variables	No. of constraints	Objective function	CPU time (min)	Optimality Status
1	1	2	3	1	2	3	2	1,078	332	78,176.63	0.017	Optimal
2	1	2	3	1	2	3	2	1,078	332	79,435.06	0.051	Optimal
3	1	2	4	1	2	4	2	1,664	584	98,855.00	0.527	Optimal
4	1	2	4	1	2	4	2	1,664	584	93,050.87	0.013	Optimal
5	1	2	4	2	2	4	2	2,088	610	88,889.88	0.666	Optimal
6	1	2	4	2	2	4	2	2,088	610	80,507.12	0.018	Optimal
7	1	2	5	2	2	4	2	2,560	802	139,897.69	0.734	Optimal
8	1	2	5	2	2	4	2	2,560	802	102,540.20	0.026	Optimal
9	2	2	5	1	3	4	3	4,470	1,227	134,532.64	5.764	Optimal
10	2	2	5	1	3	4	3	4,470	1,227	149,999.18	1.988	Optimal
11	2	3	6	1	3	6	3	8,172	2,349	177,445.68	4.833	Optimal
12	2	3	6	1	3	6	3	8,172	2,349	154,605.47	0.533	Optimal
13	2	3	6	2	4	6	3	10,608	2,436	162,710.24	11.121	Optimal
14	2	3	6	2	4	6	3	10,608	2,436	177,828.94	9.152	Optimal
15	2	3	7	2	4	8	3	14,700	3,930	175,661.49	17.238	Optimal
16	2	3	7	2	4	8	3	14,700	3,930	194,864.93	14.123	Optimal
17	2	3	8	2	5	10	4	28,020	7,928	276,904.27	29.801	Optimal
18	2	3	8	2	5	10	4	28,020	7,928	253,935.82	31.001	Optimal
19	2	4	8	3	6	12	4	42,976	9,876	270,500.67	75.371	Optimal
20	2	4	8	3	6	12	4	42,976	9,876	273,304.90	66.235	Optimal

Figure 3. The comparison between the CPU Time, and number of decision variables and constraint.

Figure 4. Schematic representation of solution obtained for Instance 9.

Figure 5. The impact of symmetry-breaking constraints on the CPU time.

Table 5. Comparison between the results of single- and multi-path settings.

	Total distance traveled (km)	Fixed cost of using vehicles	Variable cost of vehicles	Holding cost	Total cost
Single-path setting	10,787.82	28,500	274,294	2,283.10	305,077.10
Multi-path setting	10,967.53	28,500	238,242.10	3,758.57	270,500.67

Table 6. The obtained results for the original case and Case 2.

	The original case		Case 2
Period	Between $\mathcal{S}$ and $\mathcal{M}$	Between $\mathcal{M}$, $\mathcal{R}$ and $\mathcal{D}$	Between $\mathcal{S}$ and $\mathcal{M}$	Between $\mathcal{M}$, $\mathcal{R}$ and $\mathcal{D}$
Period 1	3	5	3	7
Period 2	2	5	2	6
Period 3	1	4	2	6
Period 4	1	4	2	5
Total	7	18	9	24
Variable cost of using vehicles	238,242.10		235,975.79
Fixed cost of using vehicles	28,500		37,500
Sum of the fixed and variable costs	266,742.10		273,475.79

Table 7. The obtained results for the original case and Cases 3 and 4.

Case	No. of vehicles used to satisfy the demands	Fixed cost of using vehicles	Variable cost of vehicles	Holding cost	Total costs
The original case	25	28,500	238,242.10	3,758.57	270,500.67
Case 3	21	31,500	342,044.32	2,221.43	380,265.75
Case 4	31	31,000	247,699.74	1,566.43	280,266.17

Table 8. The values of coefficients $M_1$ and $M_2$.

Coefficients	Values
$M_1$	$\underset{i,t}{max}\left(O_{it}\right)$
$M_2$	$\underset{i}{max}\left(C{F}_i\right)$

Table 9. The obtained results for the original case and Case 5.

Case	Percentage of demands satisfied over all periods (%)	Lost sale cost	Fixed cost of using vehicles	Variable cost of vehicles	Holding cost	Total costs
The original case	100	0	28,500	238,242.10	3,758.57	270,500.67
Case 5	90	24,950	23,000	207,759.31	2,105	257,859.31

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