4.1 Parameters
- \(G_{i,t,b}\) : available donors for blood type \(i\) during week \(t\) at day \(b\).
- \(D_{i,t,b}\) :demand for blood type \(i\) during week \(t\) at day \(b\).
- \(H_{i,t,b}\) : daily collected units for blood type i available during week t at day b for hospital w.
- \(M_{i}\) : minimum inventory level of available whole blood units for blood type \(i\).
- \(F_{i,t,b}\) : maximum capacity level of available whole blood units for blood type \(i\) during week \(t\) at day \(b\).
- \(L_{}\) : cost of collected whole blood unit, fixed for all blood types across all days of the week.
- \(Q_{}\) : penalty cost for expired units.
4.2 The decision making variables
- \(C_{i,t,b}\) : inventory level for blood type \(i\) during week \(t\) at day \(b\).
- \(Y_{i,t,b}\) : expired units of blood type \(i\) during week \(t\) at day \(b\).
- \(X_{i,t,b}\) : collected units for blood type \(i\) during week \(t\) at day \(b\).
4.3 The constraints
1. Resources constraint states that the number of daily collected units for any blood type shouldn't exceed the number of available units from regular donors:
\(X_{i,t,b} \leq G_{i,t,b}\)
2. Demand constraint states that the demand shouldn't exceed available resources, i.e capacity:
\(D_{i,t,b} \leq G_{i,t,b}\)
3. Inventory equilibrium constraint define how the daily inventory is updated everyday. It's set to equal collected units in the same day and any left units from previous weeks minus the demand and the identified expired units :
\(C_{i,t,b}=0.9\times X_{i,t,b}+C_{i,t-1,b}-D_{i,t,b}-Y_{i,t,b}\)
4. Minimum inventory constraint is defined to prevent shortages in inventory by always keeping the inventory above a minimum level:
\(C_{i,t,b} \leq M_{i,t,b}\)
5. Constraint of valid inventory is defined to identify expired units daily. The expired units at any point of time is a unit in inventory that was collected at the week \(t-5\) during the first day of that week :
\(Y_{i,t,b} = C_{i,t-5,1}\)
6. Inventory capacity constraint at any point is defined as the inventory level plus collected units and it should be less than the maximum capacity: