Table 1 Set, parameters and decision variables of the MILP model
From: Home care vehicle routing problem with chargeable overtime and strict and soft preference matching
Sets | |
---|---|
I | set of jobs |
C | set of clients |
K | set of caregivers |
M | set of strict preferences |
F | set of soft preferences |
H | set of time periods in the planning horizon |
W ⊂ H | subset of weekend time periods |
N ⊂ H | subset of night time periods |
Parameters | |
d_{i} | duration of job i ∈ I in time periods |
t_{i} | starting time period of job i ∈ I |
\({\theta _{i}^{c}}\) | equal to 1 if job i ∈ I belongs to client c ∈ C; 0 otherwise |
n_{i} | equal to 1 if job i ∈ I requires a night shift afterwards; 0 otherwise |
τ | regular working time (same for all caregivers k ∈ K) |
S_{k} | maximum working time of caregiver k ∈ K |
ϕ_{c} | equal to 1 if client c ∈ C is willing to pay overtime; 0 otherwise |
\({\gamma _{c}^{k}}\) | number of strict preference mismatches between client c ∈ C and caregiver k ∈ K |
\(\widetilde {\gamma }_{c}^{k}\) | number of soft preference mismatches between client c ∈ C and caregiver k ∈ K |
δ_{ij} | travel time from job i ∈ I to job j ∈ I or vice versa |
\(\widetilde {\delta }_{i}^{k}\) | travel time from depot of caregiver k ∈ K to job i ∈ I or vice versa |
ρ | minimum number of time periods a caregiver can spend on a home-based break |
β | maximum number of time periods a caregiver can spend on a break without going home |
ξ_{k} | equal to 1 if caregiver k ∈ K is willing to work on the weekend; 0 otherwise |
ν_{k} | equal to 1 if caregiver k ∈ K is willing to work at night; 0 otherwise |
λ_{k} | equal to 1 if caregiver k ∈ K is willing to perform night shifts; 0 otherwise |
D | number of time periods in a day |
Λ | big number |
α_{M} | weight of the strict preference mismatch in the objective function |
α_{F} | weight of the soft preference mismatch in the objective function |
α_{O} | weight of the overtime paid by the provider in the objective function |
α_{T} | weight of the total travel time in the objective function |
Decision variables | |
\({z^{k}_{i}}\) | equal to 1 if job i ∈ I is assigned to caregiver k ∈ K; 0 otherwise |
\(x^{k}_{ij} \) | equal to 1 if job j ∈ I is done immediately after job i ∈ I by caregiver k ∈ K |
without returning home; 0 otherwise | |
\(y^{k}_{ij} \) | equal to 1 if job j ∈ I is done immediately after job i ∈ I by caregiver k ∈ K |
when returning home between the two jobs; 0 otherwise | |
\({f_{i}^{k}}\) | equal to 1 if job i ∈ I is the first job of caregiver k ∈ K; 0 otherwise |
\({l_{i}^{k}}\) | equal to 1 if job i ∈ I is the last job of caregiver k ∈ K; 0 otherwise |
\({W^{k}_{c}}\) | overtime (positive) or undertime (negative) of caregiver k ∈ K on client c ∈ C |
\({O^{k}_{c}} \) | overtime of caregiver k ∈ K on client c ∈ C |
σ_{k} | overtime of caregiver k ∈ K not paid by the clients |
u | auxiliary binary variable |
\({v^{k}_{c}}\), \({\omega ^{k}_{c}}\) | auxiliary binary variables ∀c ∈ C,k ∈ K |
χ_{ij} | auxiliary binary variables ∀i,j ∈ I |