A Hybrid Approach to the Optimization of Multiechelon Systems

Table 4

Decision variables and constraints before after transformation.

Before transformation	After transformation	Description
Decision variables
		Transformation of decision variables level 1 from the arc model arc() to the route model ().
				Transformation of decision variables level 2 from the arc model arc() to the route model ().
Constraints
(1)		Objective function after transformation, different decision variables, the same in terms of the essence and functionality.
(2)		Number of 1-type resources (CLP-determined)
(3)	—	Supply balance equation for 1-level nodes is unnecessary after transformation. This is a result of the route model to which particular vehicles are allocated.
(4)		Number of 2-type resources (CLP-determined)
(5)	—	Vehicle balance equation for level 2 is unnecessary after transformation. This is a result of the route model to which particular vehicles are allocated.
(6)		Supply balance for satellites.
(7)		Number of tours for level 1 resulting from the capacity of vehicles.
(8)	—	Supply balance constraint for recipients is not required. In the route model, the supply volume is calculated for the route.
(9)	—	Supply volume constraint resulting from the vehicle capacity is unnecessary for level 2. The routes are generated only for the allowable capacities.
(10)	—	No return loads from satellite to depot (10). The routes are generated so as to automatically ensure this.
(11)	—	No return loads from the customer to satellite (11). The routes are generated so as to automatically ensure this.
(12), (13)	—	No variable after transformation.
(14)–(16)		No overlapping deliveries to customers.
(17)	—	This is ensured by the route model.
(18)–(20)	,	Integer and binary
(21)–(23)	—	Additional constraints are not necessary in the model with routes.