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
WH subset of weekend time periods
NH subset of night time periods
Parameters
di duration of job iI in time periods
ti starting time period of job iI
\({\theta _{i}^{c}}\) equal to 1 if job iI belongs to client cC; 0 otherwise
ni equal to 1 if job iI requires a night shift afterwards; 0 otherwise
τ regular working time (same for all caregivers kK)
Sk maximum working time of caregiver kK
ϕc equal to 1 if client cC is willing to pay overtime; 0 otherwise
\({\gamma _{c}^{k}}\) number of strict preference mismatches between client cC and caregiver kK
\(\widetilde {\gamma }_{c}^{k}\) number of soft preference mismatches between client cC and caregiver kK
δij travel time from job iI to job jI or vice versa
\(\widetilde {\delta }_{i}^{k}\) travel time from depot of caregiver kK to job iI 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
ξk equal to 1 if caregiver kK is willing to work on the weekend; 0 otherwise
νk equal to 1 if caregiver kK is willing to work at night; 0 otherwise
λk equal to 1 if caregiver kK is willing to perform night shifts; 0 otherwise
D number of time periods in a day
Λ big number
αM weight of the strict preference mismatch in the objective function
αF weight of the soft preference mismatch in the objective function
αO weight of the overtime paid by the provider in the objective function
αT weight of the total travel time in the objective function
Decision variables
\({z^{k}_{i}}\) equal to 1 if job iI is assigned to caregiver kK; 0 otherwise
\(x^{k}_{ij} \) equal to 1 if job jI is done immediately after job iI by caregiver kK
  without returning home; 0 otherwise
\(y^{k}_{ij} \) equal to 1 if job jI is done immediately after job iI by caregiver kK
  when returning home between the two jobs; 0 otherwise
\({f_{i}^{k}}\) equal to 1 if job iI is the first job of caregiver kK; 0 otherwise
\({l_{i}^{k}}\) equal to 1 if job iI is the last job of caregiver kK; 0 otherwise
\({W^{k}_{c}}\) overtime (positive) or undertime (negative) of caregiver kK on client cC
\({O^{k}_{c}} \) overtime of caregiver kK on client cC
σk overtime of caregiver kK not paid by the clients
u auxiliary binary variable
\({v^{k}_{c}}\), \({\omega ^{k}_{c}}\) auxiliary binary variables ∀cC,kK
χij auxiliary binary variables ∀i,jI