M ATERIALES Y RECURSOS DIDÁCTICOS CURSO SEXTO

We now present the results of a comprehensive numerical study of the impacts on the optimal order-up-to level and long-run average cost by the system parameters. We will discuss the results of the numerical study in the context of the following supply chain. Consider an international supply chain subject to border closures in which a domestic manufacturer orders a single product from a single foreign supplier. Orders are measured in units of container loads and are placed each day. The containers are shipped by some mode of transportation where the leadtime from the supplier to the international border of the manufacturer’s host nation is deterministically L days. The order remains at the border until the first day in which the border is open, at which point the order and all other orders arriving to, or waiting at, the border cross and immediately arrive at the manufacturer.

Table 1 displays the system parameters values that we study and Table 2 classifies the parameter combinations into 13 instances for reference purposes. The only parameter to remain constant throughout the numerical study is the per unit purchase cost, c = $150, 000. From Theorem 1, we note that the purchase cost does not affect the inventory policy. The purchase cost only contributes to the long-run average cost as a fixed cost, e.g. cE[D]. Varying its value only linearly changes the long-run average cost at rate E[D] and therefore provides no important insights about the model.

Table 1: Numerical study design (border closure model without congestion).

Parameter Values Purchase Cost, c $150,000 Holding Cost, h $100, $500 Penalty Cost, p $1,000, $2,000 Minimum Leadtime, L 1, 7, 15 Transition Probability, p_OC 0.001, 0.003, 0.01, 0.02, 0.05, 0.1, 0.2,...,0.8, 0.9, 0.95 Transition Probability, pCO 0.05, 0.1, 0.2,...,0.8, 0.9, 0.95

Demand Distribution Poisson(Mean=0.5), Poisson(Mean=1)

Given a purchase cost of $150,000, a holding cost of $100 per day represents a 24.33% annual holding cost rate. We believe that this rate is reasonable for most industries (an

Table 2: Parameter instances (border closure model without congestion).

Instance L h p Demand Dist.

1 1 $100 $1,000 Poisson(0.5) 2 1 $100 $2,000 Poisson(0.5) 3 1 $500 $1,000 Poisson(0.5) 4 1 $500 $2,000 Poisson(0.5) 5 7 $100 $1,000 Poisson(0.5) 6 7 $100 $2,000 Poisson(0.5) 7 7 $500 $1,000 Poisson(0.5) 8 7 $500 $2,000 Poisson(0.5) 9 15 $100 $1,000 Poisson(0.5) 10 15 $100 $2,000 Poisson(0.5) 11 15 $500 $1,000 Poisson(0.5) 12 15 $500 $2,000 Poisson(0.5) 13 1 $100 $1,000 Poisson(1)

annual holding cost rate of 23% is actually cited as conservative for high-technology indus- tries in [20]). While a holding cost of $500 per day is excessive in reality, its extreme value highlights the effects that the holding cost can have on the policy and long-run average cost. The penalty costs of $1,000 and $2,000 per day respectively represent an annual penalty cost of 2.4 and 4.8 times the purchase cost. Although we do not consider customer service rates in this thesis, the greater the penalty cost, the higher the corresponding customer level.

We consider minimum leadtimes, that is, the leadtime from the supplier to the border, of 1, 7, and 15 days. A minimum leadtime of 1 day corresponds to supply chains which may utilize air cargo (such as for high-technology firms) or to supply chains in which the manufacturer and supplier are located in close proximity and the supply crosses a land border. A specific example of the latter is the automotive supply chain with suppliers in Canada and manufacturers in the Midwestern United States. The Canadian automotive supplier industry is predominantly based in the province of Ontario and the majority of the supply is shipped to the United States by truck. According to one automotive industry expert, the leadtime from Canadian suppliers to the US-Canadian border crossings (for example, at Niagara Falls, Windsor, and Sarnia) is typically less than one day. Even for supply that is shipped by rail, a reasonable estimate for the leadtime from supplier to

the border is one to two days (for example, stamped parts shipped from General Motor’s Oshawa complex in Canada to the United States). Since many of these manufacturing plants are located in the Midwestern United States, the leadtime from the US-Canadian border to the manufacturing facility can be considered negligible. A minimum leadtime of 7 days corresponds to a supply chain system with either longer physical travel time to the border or one that requires production time between order receipt and ship date. Finally, a minimum leadtime of 15 days corresponds to typical ocean carrier services from Asia to the Western United States. With growing Asian economies and increased outsourcing to Asia, this transportation mode and route are increasingly important to the world economy.

We study a wide range of border state transition probabilities in order gain a broad perspective of how the border system affects the optimal order-up-to level and long-run average cost. The transition probabilities correspond to an expected inter-closure time

ranging from approximately three years to one day (respectively, p_OC = 0.001 and p_OC =

0.95) and an expected closure time ranging from 20 days to 1 day (respectively, pCO = 0.05

and pCO= 0.95). In reality, the expected inter-closure times are large and not on the order

of days. In the comprehensive numerical study of border closure model with congestion in Chapter 4, we consider a restricted subset of realistic transition probabilities. For both models, we believe that the range of expected closure times represent realistic possibilities. We consider per period demands that are approximately identically and independently distributed Poisson random variables with means of 0.5 and 1 units per day. We use a

truncated Poisson distribution with a maximum realizable demand in any period of d_M

units. A truncated Poisson distribution assigns Poisson probabilities to all demand real-

izations up through dM −1 and a probability of 1 − G(dM −1) to dM where dM is chosen

such that 1 − G(d_{M −1}) < ² for some ² > 0. For example when the mean demand is

0.5 containers per day, the maximum realizable demand in any period is d_M = 10 and

P (D = 10) = 1 − G(9) = 1.63 × 10−10_.

We now present the results of the numerical study. Results are depicted in the figures and the corresponding numerical values are presented in tables in Appendix A.