Extended model with lead time variability

5.2.1 Incorporating the lead time factor

In the base model, we assume a deterministic lead time in the supply of aluminum cans to the brewery. We considered that, irrespective of the order quantity, the can supplier completes the lot production in exactly four weeks and ships the produced empty cans to the brewery at the beginning of the fifth week. In this model extension, we incorporate uncertainty in the lead time by considering a stochastic lead time with a discrete probability distribution.

5.2.1.1 Variability of the lead time

We consider two levels of lead time variability that we denote as high and low. The high variability corresponds to the case to which we assign probabilities of 0.464, 0.214, 0.179 and 0.143 to lead time durations of 3.5, 4, 4.5 and 5 weeks, respectively. The low variability corresponds to the case in which we assign the same probabilities to lead time durations of 3.75, 4, 4.25 and 4.5 weeks, respectively. With these discrete probability distributions the average lead time duration in both cases is four weeks, the same duration used in the base model. This facilitates the comparison between the base and the extended models. The standard deviations of the discrete probability distributions with high and low variability are 0.551 and 0.275 weeks, respectively.

5.2.1.2 Modeling the stochastic lead time duration

To introduce the variability of the lead time duration into our model, we make a number of assumptions that are specific to the conceptual background of our model. These assumptions are necessary to justify our calculations of the relevant costs. First, we assume that the cans are produced in a continuous process with a production rate that is constant for each batch of cans ordered for the brewery’s consumption in a specific week. Second, the variability of the lead time duration stems from a change in this production rate (for example due to increased capacity allocation for another customer), and not from a delay in the production start time or a disruption in the production process. This assumption allows us to determine the quantity of cans produced at the end of the four weeks duration (the expected completion date) in a proportional manner as we will discuss in the following section. Third, we assume that shipments from the can supplier to the brewery are made once a week. This means that any unfinished portions of an order will be shipped with the next week’s order. Finally, we assume that no early shipments are allowed. This means that a batch of cans that is completed earlier than the expected delivery date remains at the supplier’s premises until the agreed shipping date.

5.2.1.3 Impact of lead time variability on the SCRM process

The introduction of a stochastic lead time duration has an impact on the production schedule and product flows discussed in the base model in which the quantity of aluminum cans shipped to the brewery (Qc) is equal to the quantity of a production lot

that started four weeks earlier (Pc). That is, in the base model, the planned production

quantity (Pc) is actually produced and the planned shipment quantity of cans (Qc) is thus

(Pc)Actual, may be less than the planned production quantity, Pc, due to a lead time duration

longer than four weeks. Consequently, the actual quantity of cans shipped to the warehouse may be less than the planned quantity (Qc). Under this new situation, the

brewery places an order with the can supplier for a quantity of cans (Qcj) that needs to be

received at the beginning of week wj. The can supplier starts producing this planned

quantity, Pc, four weeks before the expected delivery time. In the event that the lead time

duration, represented by X, is longer than the expected four weeks, only a proportion of the ordered quantity would be ready for shipment. This proportion is equal to Pc x 4/X.

The remaining balance that is still in production is shipped with the batch produced for the next week wj+1. In the event that the lead time duration is shorter than four weeks, the

supplier holds all the produced quantity and delivers it, as scheduled, at the beginning of week wj.

5.2.2 Modifications to the base model formulation

As explained above, incorporating a stochastic lead time duration into the model makes the quantity actually produced by the can supplier every week a variable quantity that is not necessarily equal to the planned production lot (Pc). When the lead time is longer than

four weeks, a proportion of Pc is produced on time while the remaining balance is still

under production, and is shipped when completed the next week. Subsequently, a holding cost for the remaining balance is added to the total opportunity cost calculations. Accordingly, equation (7) in the base model is modified. The first term in this formulation represents the present value of the holding cost associated with carrying the surplus quantity of aluminum cans during the production phase for the whole lead time period. The surplus is determined by the weekly ending inventory. In other words, this

holding cost is the cost of insurance against uncertain demand. We only include the carrying cost corresponding to this surplus quantity, and not to the whole production lot, to be consistent with our definition of the opportunity cost. All the components of the opportunity cost penalize the supply chain for the deviations from ‘perfect’ decisions. Such decisions can be made only if ’perfect’ information on demand quantity and aluminum price is known a priori, which, of course, can never be the case in reality. In accordance with the opportunity cost concept that we adopt, we add to the cost in equation (7) the holding cost corresponding to the proportion of the production quantity that is delayed due to a longer lead time. This cost is perceived as the cost of insurance against uncertain lead time in supplying the cans. To determine this cost, we compute for every production week the actual quantity produced and, correspondingly, the remaining balance quantity still in production. These quantities are computed as follows.

4 L~ if L ~4 P 4 L ~ if P ) (P c c c c c Actual c (i)

The balance in production, BIP = Pc – (Pc)Actual (ii)

Substituting the value of (Pc)Actual in (ii) by the relevant values from (i), the balance in

production is determined as follows.

4 L ~ if L ~4 1 P 4 L ~ if 0 BIP c c c c (iii)

As justified above, the cost of carrying this balance for four weeks is added to the cost of carrying the quantity produced in surplus. These two costs are calculated in (7a).

j) r(T - c1 1 c0 0 8 1 c cj 4) (j c ,0 (u h u h ) 4e 0 L~ 4 - 1 Max P E j (7a) 13 5 j j) r(T - c1 1 c0 0 cj(u h u h )e 0 E (7b)

The cost in (7b) corresponds to the cost of carrying the surplus quantity of cans in the warehouse. The cost of carrying inventory in a downstream location along the supply chain would be higher than the cost of carrying the same inventory in an upstream location.