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 | |
| di | duration of job i ∈ I in time periods |
| ti | starting time period of job i ∈ I |
| \({\theta _{i}^{c}}\) | equal to 1 if job i ∈ I belongs to client c ∈ C; 0 otherwise |
| ni | equal to 1 if job i ∈ I requires a night shift afterwards; 0 otherwise |
| τ | regular working time (same for all caregivers k ∈ K) |
| Sk | maximum working time of caregiver k ∈ K |
| ϕc | 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 |
| δij | 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 |
| ξk | equal to 1 if caregiver k ∈ K is willing to work on the weekend; 0 otherwise |
| νk | equal to 1 if caregiver k ∈ K is willing to work at night; 0 otherwise |
| λk | 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 |
| 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 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 |
| σk | 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 |
| χij | auxiliary binary variables ∀i,j ∈ I |