Supplier model formulation

First of all, it should be noted that in the supplier model, under the assumptions made, fSC(t); t = 0; :::; T 1g are input variables (parameters), given by:

SC(t) = O(t) _{8t j t = 0; : : : T + 1} (6.12) SC( 1) = 0; SC(T ) = 0; SC(T + 1) = 0 (6.13)

Equation (6.12) links the customer model presented in subsection 6.4.4 and the supplier model we introduce in the current subsection.

Objective: Minimize total undiscounted relevant costs over the planning horizon T Objective function: CaN + T +1_X t= 1 CfF (t)+ T +1 X t= 1 fCosYos(t) + CtscSC(t) + CtcsCS(t)g+ T +1_X t= 1 fChesIes(t) + Chf sIf s(t)g (6.14) Non-negative conditions of the variables and special periods values:

F (t) 0; _{8t j t = 1; : : : T} 2 (6.15) Yos(t) 2 f0; 1g ; 8t j t = 0; : : : T (6.16) CS(t) 0; _{8t j t = 2; : : : T} (6.17) Ies(t) 0; 8t j t = 0; : : : T + 1 (6.18) If s(t) 0; 8t j t = 0; : : : T + 1 (6.19) Iec(t) 0; 8t j t = 0; : : : T (6.20) If c(t) 0; 8t j t = 0; : : : T (6.21)

F (T 1) = 0; F (T ) = 0; F (T + 1) = 0 (6.22) Yos( 1) = 0; Yos(T + 1) = 0 (6.23) CS( 1) = 0; CS(0) = 0; CS(1) = 0 (6.24) Iec(T + 1) = 0; If c(T + 1) = 0 (6.25) If s( 1) = 0; Ies( 1) = N; If c( 1) = 0; Iec( 1) = 0; (6.26) Filling by supplier: F ( 1) SC(0) (6.27) F (t) Ies(t); 8t j t = 1; 0; 1; : : : T 2 (6.28)

Ordering and transport:

CS(t) = SC(t 1) _{8t j t = 2; : : : T} (6.29) SC(t) 104Yos(t) 8t j t = 0; : : : T 1 (6.30) CS(T + 1) = Iec(T ) + If c(T ) (6.31) CS(T + 1) 104Yos(T ) (6.32) Inventory: SC(t) If s(t 1) + F (t 1) 8t j t = 0; : : : T 1 (6.33) Mass balances:

If s(t) = If s(t 1) + F (t 1) SC(t) 8t j t = 0; : : : T + 1 (6.34)

Ies(t) = Ies(t 1) F (t 1) + CS(t 1) 8t j t = 0; : : : T + 1 (6.35)

If c(t) = If c(t 1) D(t 1) + SC(t 1) 8t j t = 0; : : : T (6.36)

Iec(t) = Iec(t 1) + D(t 1) CS(t) 8t j t = 0; : : : T (6.37)

Equation (6.14) shows the objective function, where the …rst term indicates the ‡eet acqui- sition cost (to be paid only once, at the beginning of the planning horizon), the second term denotes the …lling cost and the third term re‡ects the transport cost, which decomposes in other three subterms: the …rst subterm indicates the …xed transport cost, to be paid every time a shipment to the customer is lauched, whereas the second and third subterm re‡ect the variable transport costs in each trip leg (from supplier to customer and viceversa). Finally, the fourth term in the objective function represents the stock keeping costs related to empty and full containers at the supplier.

The …rst block of constraints (equations (6.15) to (6.26)) show the non-negative conditions of the variables, as well as the …xed values taken by the variables in the special periods. Customer demand spans from period t = 0 to period t = T , so the last order is placed in period T 1. Therefore, the last period with …lling is period T 2, so zero containers will be …lled in periods T 1; T;and T + 1 (constraints (6.22)).In addition, orders of full containers are (potentially) shipped to the customer in time periods spanning from t = 0 to t = T 1. In t = T , the special trip for retrieving all containers at the customer is launched. Therefore, in periods t = 1 and t = T + 1 it is not possible to launch a shipment (constraints (6.23)).

Constraints (6.24) and (6.17) are a consequence of the fact that the …rst retrieval of empty containers at the customer (equal exchange recovery strategy) takes place at the beginning of period t = 2. Hence, CS(t) is a variable from period 2 onwards (constraint (6.17)). The values taken by CS(t) for t = 1, t = 0 and t = 1 are provided in (6.24). Constraints (6.25)

force to retrieve all remaining containers at the customer at the end of the planning horizon. Constraints (6.26) provide the set up values of the inventory variables.

Next, we have the blocks of real constraints. Constraints (6.27) and (6.28) refer to the …lling process: as all demand has to be ful…lled immediately, the quantity …lled in the …rst period has to cover at least the demand re‡ected in the …rst order (constraint (6.27)); in addition, the quantity of containers to be …lled in a period is limited by the on hand inventory of empty containers avaliable at the supplier in that period (6.28). This constraint applies to all periods where there is a (potential) …lling process.

Constraints from (6.29) to (6.32) refer to the ordering and transport process. Constraint (6.29) ensures the application of the equal exchange policy, so it is only e¤ective from period 2 onwards. Constraint (6.30) is used to activate the …xed transport cost to be paid every time a shipment is launched from supplier to customer (and back to the supplier). The restriction spans all the periods in which potentially a "regular" shipment can take place (from 0 to T 1). In period T , a special shipment is lauched from the supplier in order to retrieve all the remaining stock of full and empty containers at the customer at the end of the planning horizon. This fact is re‡ected in constraints (6.31) and (6.32) , that result from the linearization of equation (*):

CS(T + 1) = Yos(T ) [Iec(T + 1) + If c(T + 1)] (*)

This equation can be transformed into two constraints:

on the one hand, CS(T + 1) 104Yos(T ) () constraint (6.31).

and on the other hand, CS(T + 1) = Iec(T + 1) + If c(T + 1) () constraint (6.32).

It should be noted that the …nal formulation proposed in (6.32) stems from the following reasoning:

CS(T + 1) = Iec0(T ) + If c0(T ) = Iec(T ) + D(T ) + If c(T ) D(T ) = Iec(T ) + If c(T )

| {z }

+

CS(T + 1) = Iec(T ) + If c(T )

An interesting result highlighted by this equation is that the total inventory at the customer (sum of full and empty containers) at the beginning of a period equals the total inventory at the end of the period. Although the sum remains constant throughout the period, the proportion of full and empties changes depending on the value of demand D(t). This result applies to customer inventory for all periods. In the particular case of the end of the planning horizon, the total inventory to be retrieved at the beginning of period t = T + 1 (CS(T + 1)) is the total inventory at the end of the precedent period (Iec0(T ) + If c0(T )) due to the fact that in instant

i = T there are no new deliveries of containers.

The next block is constraint (6.33), which represents the inventory limit of the system: the maximum amount that can be shipped in each period to the customer is the available on hand inventory of full containers at the beginning of the period. This restriction spans to all the periods where there is (potentially) a shipment.

Finally contraints (6.34) to (6.37) represent the mass balances between empty and full containers in both the customer and the supplier. The inventories of full and empty containers at the supplier (Constraints (6.34) and (6.35)) take into account the …lling process and the shipment from and to the customer. They span until period T + 1 because in the last period the inventory at the supplier is still updated with the last shipment retrieving all the remaining containers at the customer. The inventories at the customer (Constraints (6.36) and (6.37)) take into account product consumption and shipments from and to supplier. In the …nal period T + 1, after the retrieval of all remaining inventories, the on hand stock of full and empty containers at the customer should be 0.