Table A.1.

Nomenclature; list of sets, indices, parameters, and decision variables.

Symbol	Definition	Symbol	Definition
$.EC$	Suffix for excessive capacity of related variable;	$.ID$	Suffix for location ID of population of related variable;
$.Pop$	Suffix for population of related variable;	$.Pos.SC$	Position (chromosome) of particle in a shortage strategy;
$.Quality$	The quality (fitness) of the particle (individual);	$.R$	Suffix for the initial results;
$.TDist$	Suffix for total distance of related variable;	$.Tpop$	Suffix for total population of related variable;
$\alpha$	Probability rate of Monte Carlo selection process;	$AA$	Set of assigned demand points (equal to $X$ decision variable of Assignement model);
$AD$	Set of Average distance of demands and allocated $VCs$ ;	$ALT.ID$	Set of IDs of the available $VCs$ from the current position of truck $v$;
$ALT.T$	Set of travel times to available $VCs$ from the current position of truck $v$;	$ASS$	Demand assignment of day $d$ to $VCs$;
$A{v}_v$	Truck availability for next destination; binary parameter;	$AVC$	Set of numbers of active vaccine centres;
$BASS$	Balanced capacity assignment of the service centres ;	$BC$	Maximum movements in staff re-allocation;
$BD$	Balanced distribution;	$Bofparticle$	The particle with best quality (fitness) in the last set of particles;
$Bofswarm$	The particle with overall best quality (fitness);	$C$	Assigned capacity of VCs in the $AS{S}_d$ for simulation;
$C_i$	Allocated capacity to the centre $i$;	$C1$	A random coefficient number in Delta in Alg. XI;
$CC$	Set of Original capacities of the $VC$;	$CCT$	Current capacity of vaccine distribution ;
$CD$	Capacity difference;	$Chr$	Chromosome;
$Clients$	Set of numbers (ID) of the pre allocated Demands to a centre;	$Cos{t}_{ij}$	The distance between centre $i$ to centre $j$;
$CR{S}_d$	Couriers distribution of day $d$ within time window $tw$;	$Currentcost$	Unchanged cost in process of the Alg. IX;
$d$	Counter for vaccination day;	$d_{ij}$	The Euclidean distance between centre $j$ and demand $i$;
$D_s$	Set of coordinates of Demand point $s$;	$D1$	A random coefficient number in $V_{current}$ in Alg. XI;
$DCD$	Dynamic capacity difference;	$Delta$	Distance of particle from Bofparticle and Bofswarm;
$DEV$	Difference of centres’ capacity from constraint;	$Dif{f}_i$	Remaining capacity of $VCs$$i$;
$DIS$	Set of distances from allocated demand points to a centre;	$Dist$	Distance between the demand and $VCs$;
$Distmatri{x}_{ij}$	The Euclidean distance between centre $j$ and demand $i$;	$DM$	Decision maker;
$DPo{p}_i$	Population of the demand point $i$ (in each day $d$) ;	$DST$	ID number of the next destination of truck $v$ in day $d$;
$DT$	Central vaccine storage place (depot);	$DV$	Delivered vaccines to $VCs$ in day $d$;
$Elite$	Individual with the least amount of fitfuncion;	$ET$	Expiration time of vaccine packages;
$Excess$	Excessive load;	$FD$	Set of locations’ IDs with no unmet demand;
$fit$	Fitness ;	$g$	Counter for offsprings of the crossover;
$j$, $j^{\prime }$	index of demand points;	$IDC$	ID number of $VCs$;
$IFC$	Individual for crossover;	$IFM1$	Selected particles for muting by mutation 1;
$IFM2$	Selected particles for muting by mutation 2;	$IND$	Set of IDs (numbers) of centre corresponding the allocating demands point;
$int$	Interval gap for receiving second dose of vaccine;	$k$	Counter for demands for the assignment model;
$L_v$	Load of vehicle $v$;	$Local$	Prefix for local variables (Set of demand points temporary allocated to $VCs$);
$LV$	Levelled capacity of $VC$;	$ℳ$	Set of IDs (numbers) of the $VCs$;
$MC$	Maximum capacity of $VC$;	$Mutant1$	The individual that has been muted by mutation 1;
$Mutant2$	The individual that has been muted by mutation 2;	$N$	Set pf Population of IDs of demands for assignment model;
$\mathcal{N}$	Set of IDs (numbers) of the Demand points;	$nc$	Number parent particles for crossover;
$NG$	Next generation of results;	$nIteration$	Number of iteration for the GA;
$NM$	Nominated $VCs$ for serving demand;	$nm1$	Number of individuals to get mutated by the mutation 1;
$nm2$	Number of individuals to get mutated by the mutation 2;	$NoT$	Number of trucks in day $d$ to distribute vaccines;
$npop$	Population of initial results in HGA Alg.s;	$OC$	Overall capacity of $VCs$;
$Offs$	$VCs$ with zero overall capacity;	$Offspring1$	The first child of parent 1 and parent 2;
$Offspring2$	The second child of parent 1 and parent 2;	$OP$	Over populated centres;
$P_{New}$	New received package;	$P_d$	Package of vaccines of day $d$;
$P_r$	Reserved vaccines;	$Parent1$	The individual that is a parent for the crossover;
$Parent2$	Another individual for parenting in the crossover;	$Particl{e}_i$	The particle I in the swarm;
$pc$	Chance of being parent for the individuals in a generation;	$PDCD$	Peak dynamic capacity difference;
$pm1$	Mutation probability rate 1;	$pm2$	Mutation probability rate 2;
$Po{p}_i$	The individual $i$ in the generation;	$Prio$	Priority ranking of mesh block area centroid;
$Queue$	Set of demand points’ IDs waiting for vaccination of the day;	$R$	Distance between all the demands and their nearest centre;
$Ratio$	Accuracy measure index;	$RC$	Remaining capacity;
$R_s$	Euclidean distance between demand and nearest centre;	$S$	The number of supply for the assignment model;
$s$	A counter for centres;	$S_{i}Seq$	Set of vaccine distribution sequences;
$Scor{e}_k$	Chance of selecting centre $k$ in the chromosome;	$Shortage$	Shortage load;
$S{Q}_v$	Set of sequence of truck $v$ visiting $VCs$;	$S{R}_w$	Surplus population in overpopulation $VCs$ of $w$(iteration);
$ST1$	Set of demands’ numbers as Service takers (first dose vaccination);	$ST2$	Set of demands’ numbers as Service takers (second dose vaccination);
$t$	A counter for demand points;	$TCC$	Total centres capacity;
$TDC$	Total capacity difference;	$TMC$	Total maximum capacity of $VCs$;
$TPD1$	Top priority demand layer 1 (first dose vaccination);	$TPD2$	Top priority demand layer 2 (second dose vaccination);
$TS$	Vehicle (truck) capacity;	$T{T}_v$	Travel time of truck $v$ from current $VC$ to the next $VC$;
$TW$	Time window for vaccine delivery;	$U$	A Boolean variable to terminate the loop in the Alg. IX;
$UC$	Undelivered vaccine demand of $VCs$;	$UC$	$OC$ that can change during the process of Alg. IX;
$UF1$	Set of not working with full of excess capacity centres;	$v$	Index of couriers trucks;
$v$	A counter for the trucks;	$V_{c}urrent$	current speed for changing;
$V_{p}ast$	Speed of changing in the last generation;	$V{C}_k.Tdist$	Total distance of the allocated demands to the centre $k$;
$V{C}_k.Tpop$	Total population of the allocated demands to the centre $k$;	$VCl$	Vaccine centre’s load;
$VDCD$	Valley dynamic capacity difference;	$VR$	vehicle sequence with minimum travel time;
$W_j$	Population of the assigned demand point $i$;	$Winner$	Index of the centre that has been selected for muting;
$WLD{2}_d$	Set of number of demands as waiting list (second dose vaccination) on day $d$;	$X_{ij}$	Allocated population of point $j$ to centre $i$;
$X_k$	Decision variable of the assignment model;	$XX$	Sum of chromosomes with less excess amount;
$Y.d1$	Total maximum daily required demand for dose 1;	$Y.d2$	Total maximum daily required demand for dose 2;
$YY$	Sum of chromosomes with less shortage amount;	$z$	A counter for crossover operation;
$ZMB$	Number of allowed set of destinations collection;	$ZT$	Vaccine expiration threshold;