V: Set of nodes. 0 is the depot (DC); others are customers (retailers).

V ' : V ' = V \ { 0 } .

i, j: Indices of nodes. i , j ∈ V . .

p: Index of the product. P is the set of all products. p ∈ P .

k, h: Indices of the vehicle. K is the set of all vehicles. k , h ∈ K .

t: Index of the time period. T is the set of all time periods. t ∈ T .

h i p : Unit holding cost of product p at node i. It is constant throughout the planning time horizon.

L i p t : Inventory level of product p at node i in the end of time t.

R p t : The available quantity of product p from suppliers at depot 0 in the beginning of time t.

D i p t : The demand of product p at node i in the end of time t.

R C i : Storage capacity of node i. It is shared by all products.

V C k : Vehicle capacity.

C i j : Unit distance cost from node i to node j.

M: A extremely big number.

FC: Fixed cost of vehicle used.

S i p k t : Shipping quantity of product p of node i by vehicle k in time t.

T K S i k t : Total shipping quantity of node i by vehicle k in time t.

T S i t : Total shipping quantity of node i in time t.

W k t : Binary variable. It equals 1 if and only if vehicle k is used in time t, 0 otherwise.

U i j k t : The flow of vehicle k traveling from node i to node j in time t.

T K S i k t : Fraction of the shipping quantity of node i by vehicle k in time t.

X i j k t : Binary variable. It equals 1 if and only if vehicle k from node i to node j in time t, 0 otherwise.

Minimize

∑ i ∈ V ∑ p ∈ P ∑ t ∈ T H C i p × L i p t + ∑ i ∈ V ∑ j ∈ V ∑ k ∈ K ∑ t ∈ T C i j × X i j k t + ∑ k ∈ K ∑ t ∈ T W k t × F C (1)

L 0 p t = L 0 p ( t − 1 ) + R p t − ∑ i ∈ V ' ∑ k ∈ K S i p k t ∀ p ∈ P , t ∈ T > 0 (2)

L i p t = L i p ( t − 1 ) + ∑ k ∈ K S i p k t − D i p t ∀ i ∈ V ' , ∀ p ∈ P , t ∈ T > 0 (3)

∑ p ∈ P L i p t ≤ R C i ∀ i ∈ V , t ∈ T (4)

∑ p ∈ P ∑ k ∈ K S i p k t ≤ R C i − ∑ p ∈ P L i p ( t − 1 ) ∀ i ∈ V ' , t ∈ T > 0 (5)

∑ p ∈ P S i p k t = T K S i k t ∀ i ∈ V ' , ∀ k ∈ K , t ∈ T (6)

∑ p ∈ P T K S i k t = T S i t ∀ i ∈ V ' , t ∈ T (7)

∑ j ∈ V X i j k t − ∑ j ∈ V X j i k t = 0 ∀ i ∈ V ' , ∀ k ∈ K , t ∈ T (8)

∑ k ∈ K F i k t = 1 ∀ i ∈ V ' , t ∈ T (9)

∑ i ∈ V ' T S i t × F i k t ≤ V C k ∀ k ∈ K , t ∈ T (10)

∑ j ∈ V X j i k t ≥ F i k t ∀ i ∈ V ' , ∀ k ∈ K , t ∈ T (11)

∑ j ∈ V U j i k t − ∑ j ∈ V U i j k t = T K S i k t ∀ i ∈ V ' , ∀ k ∈ K , t ∈ T (12)

∑ j ∈ V U j i k t − ∑ j ∈ V U i j k t = T S i t × F i k t ∀ i ∈ V ' , ∀ k ∈ K , t ∈ T (13)

U i j k t ≤ X i j k t × V C k ∀ i , j ∈ V , ∀ k ∈ K , t ∈ T (14)

∑ i ∈ V ∑ j ∈ V X i j k t ≤ M × W k t ∀ k ∈ K , t ∈ T (15)

W h t ≤ W k t ∀ h , k ( h > k ) ∈ K , t ∈ T (16)

U i 0 k t = 0 ∀ i ∈ V ' , ∀ k ∈ K , t ∈ T (17)

X i i k t = 0 ∀ i ∈ V , ∀ k ∈ K , t ∈ T (18)

L i p t ≥ 0 ∀ i ∈ V , ∀ p ∈ P , t ∈ T (19)

S i p k t , T K S i k t , T S i t , D i p t ≥ 0 ∀ i ∈ V ' , ∀ p ∈ P , ∀ k ∈ K , t ∈ T (20)

W k t , X i j k t = { 0 , 1 } ∀ i , j ∈ V , ∀ k ∈ K , t ∈ T (21)

The objective function (1) minimizes the total cost, including inventory cost, traveling distance cost and vehicle using cost. Constraints (2) and (3) ensure the inventory balance at depot and customers while constraints (4) define the maximum inventory capacity. Constraints (5) state the total shipping quantity of each customer is not over the remaining inventory capacity. Constraints (6) and (7) link the shipping quantity to the vehicle routing variables. Constraints (8) ensure the same vehicle arrives and leaves the customer served. Constraints (9) make sure each customer served by at least one vehicle receives its full planned shipment in each time period. Constraints (10) limit the delivery quantity to vehicle capacity while constraints (11) permit each customer to be served by more than one vehicle. Constraints (12) and (13) are the flow balance constraints and eliminate the sub-tours. Constraints (14) ensure the flow is not over the vehicle capacity when traveling between two nodes. Constraints (15) state the customers can be served only by the vehicle which is used. Constraints (16) say the vehicle using order. Constraints (17) ensure there is no product when the vehicle returns to the depot. Constraints (18) prohibit the same vehicle back to the same customer served. Constraints (19) and (20) state the non-negative integer variables while constraints (21) define the binary variables.