A Hybrid Method for Modeling and Solving Supply Chain Optimization Problems with Soft and Logical Constraints

Table 3

(a) Decision variables used in the MILP and MILP_T models. (b) Constraints used in the MILP and MILP_T models.

(a)

MILP	MILP_T	Description of the decision variables after the multidimensional transformation
()	()
		Decision variable , unlike the initial decision variables , , is generated only for technologically possible indices combinations; it defines the allocation size of product to the route of deliveries
			Unnecessary	After transformation replaced by the appropriate factor for the route, generated by the CLP
		Without change, the same sense
	Unnecessary	After transformation replaced by the appropriate factor for the route, generated by the CLP
		Without change, the same sense
		Without change, the same sense

(b)

MILP	MILP_T	Description of the constraints after the multidimensional transformation
()	()
(1)		Objective function (the same meaning)
(2)		The manufacturer’s production capacity (the same meaning)
(3)		The customer’s demands (the same meaning)
(4)	Unnecessary	Redundant after transformation (results from the decision variable )
(5)		The distributor’s capacity/volume (the same meaning)
(6)	Unnecessary	Redundant after transformation; During the transformation generates routes that meet the time constraint
(7), (8), (9), (10)	, , ,	The appropriate number of means of transport (the same sense)
(11)		The sum of the number of means of transport used does not exceed the limit of their number (the same meaning)
(12)	Unnecessary	Redundant after transformation (ensures constraint (4))
(13), (14)	Unnecessary	Redundant after transformation; After transformation replaced by the appropriate factor for the route
(15), (16)	Unnecessary	Redundant after transformation; Calculate the auxiliary parameters, performed by CLP in transformation
(17)–(27)	(11)–(17)	Integrity and binarity
(28)–(30)	–	Logic constraints (the same meaning)
—	(10)	Resetting the nonexisting routes after transformation
	()
—	–	Additional constraints increase the range of propagation