A metaheuristic to support the distribution of COVID-19 vaccines

A metaheuristic is developed based on the combination and adaptation of GRASP (Greedy Randomized Adaptive Search Procedure) with VND (Variable Neighborhood Descent), considering different refinement operators. Finally, as a way of validating the model, a numerical application in the state of Pernambuco (Brazil) was performed.

Main findings

Metaheuristic proved to be effective for developing adequate planning for the allocation of ampoules with vaccines to combat COVID-19. Effective analysis was obtained in the evaluation of the proposed algorithm, both in terms of computational effort and the quality of the final solution. An efficiency of approximately 75% was obtained in relation to the current distribution procedure adopted by the state of Pernambuco.

Implications for theory and practice

To mitigate disease, adequate logistics for transporting and distributing vaccines is essential, especially in emergency situations to face pandemic crises. Thus, the developed metaheuristic can support governments and companies in any situation demanded, making the decision of how the distribution of the ampoules will be more agile.

Keywords
COVID-19; Vehicle routing; GRASP; VND

Procedure GRASP (f (.), g (.), N (.), GRASPmax, s)
1: f* ← ∞;
2: For (Iter = 1, ..., GRASPmax):
3: Construction (g (.), α, s);
4: LocalSearch (f (.), N (.), s);
5: If (f (s) < f): 6: s ← s; 7: f* ← f (s); 8: End-If; 9: End-For; 10:s ← s*;
11: Return s;
End GRASP

Procedure Construction (g (.), α, s)
1: s ← ∅;
2: Initialize the candidate set C;
3: While (C ≠ ∅):
4: g ( $t_{m i n}$ ) = min{g(t) \| t ∈ C};
5: g ( $t_{m a x}$ ) = max{g(t) \| t ∈ C}; 6: LCR = {t ∈ C \| g(t) ≤ g ( $t_{m i n}$ ) + α (g ( $t_{m a x}$ ) - g ( $t_{m i n}$ ))}; 7: Select at random one element t ∈ LCR; 8: s ← s ∪ {t}; 9: Update the candidate set C; 10:End-While;
11:Return s;
End Construction

Procedure LocalSearch (f (.), N (.), s)
1: V = {s’ ∈ N(s) \| f (s’) < f(s)};
2: While (\|V\| > 0):
3: Select s’ ∈ V;
4: s ← s’;
5: V = {s’ ∈ N(s) \| f(s’) < f(s)}; 6: End-While; 7: Return s;
End LocalSearch

Procedure VND (s, Kmax)
1: Initialization: s = Initial Solution Generated; Define a Set of Neighborhood Structures $N_{k}$ , k = 1, 2, ..., Kmax;
2: While (No improvement is found):
3: k=1;
4: While (k ≠ Kmax);
5: x’ = BestNeighbor ( $N_{k}$ (x)); 6: If (x’ better than x): 7: x=x’; 8: Else: 9: k=k+1; 11: End-If; 12: End-While; 13: End-While; 14: Return s;
End VND

Algorithm GRASP + VND (Matrix of distances, demands, capacity, α, itermax)
1: s = $\emptyset$
2: $f_{b e s t}$ = 0
3: For (iter = 1, ..., itermax):
4: For (each vehicle):
5: Initialize a list of candidates C
6: While (C $\neq \emptyset$ ):
7: $g (t_{m i n}) =$ min $\{g (t) \| t \in C\}$
8: $g (t_{m a x}) = m a x \{g (t) \| t \in C\}$
9: RCL = $\{t \in C \| g (t) \leq g (t_{m i n}) + α (g (t_{m a x}) - g (t_{m i n}))\}$
10: Select one of the nodes that belongs to list C at random ( $t \in$ RCL)
11: s = s $\cup$ {t}
12: Update list C 12: End-While; 13: End-For;
14: k = 1
15: While (k ≤ neighborhood operator):
16: Search for the best neighbor $s^{'} \in N^{(k)}$
17: if (f( $s^{'}$ ) > f( $s_{b e s t}$ )):
18: s = $s^{'}$
19: k = 1
20: else:
21: k = k + 1 22: End-If; 23: End-While;
24: if (f(s) > $f_{b e s t}$ ):
25: $s_{b e s t}$ = s
26: $f_{b e s t}$ = f(s) 27: End-If; 28: End-For;
29: Return s

	Case	Real	1	2	3
Distance (km)	Best	14,434	3,974	3,513	3,655
	Average	-	4,320.55	3,704.43	3,662.27
	Standard deviation	-	211.06	94.01	8.48

Time (s)	Best	0.001	1,050.36	206.78	22.26
	Average	-	2,870.11	384.49	37.35
	Standard deviation	-	971.65	108.36	9.94

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