Transportation & Supply Chain Costs: Answers to Questions John H. Vande Vate Spring, 2001 1

Assignment #1 (Group) How does the average inventory at the origin compare with the common estimate: Average Shipment Size/2 or Production Rate * Average Headway/2? 2

Analysis Item Hours from one headway Total Item Hours Total Time headway*shipment size/2 headway*(headway*Rate of Demand)/2 headway2*Rate of Demand/2 Total Item Hours sum (hi2*D’)/2 Total Time sum hi Average Items in Inventory sum (hi2*D’)/[2*sum(hi)] 3

Production Rate * Average Headway/2? Average Inventory sum (hi2*D’)/[2*sum(hi)] How does this compare with Production Rate * Average Headway/2? i.e., with Avg(h)*D’/2? How does sum(hi2)/sum(hi) compare with Avg(h)? 4

How does sum(hi2)/sum(hi) compare with Avg(h)? Some extremes There’s just one headway Headways are 0 and H More general (h12 + h22)/(h1+h2) ? (h1+h2)/2 2(h12 + h22) ? (h1+h2)2 (h1-h2)2 ? 0 5

Conclusion If there is variation in the headways then the average number of items in inventory is greater than estimated from the average headway Why? 6

(Un)Common Sense Inventories associated with longer headways are not only larger, they are also around longer. 7

Apparent Contradiction Time Cumulative number of items Cumulative Production Cumulative Demand In transit Apparent Contradiction “The maximum accumulation at the destination is the same as it is at the origin” Max. Inv. at Destination Max. Inv. at Origin 8

Assignment #2 Explain this apparent contradiction. Is Daganzo just plain wrong? Can we calculate the maximum inventory at the destination? What about the average inventory? 9

Explanation Daganzo is right is we also assume that Shipment arrives when customer inventory reaches 0 What does this say about transit times, etc.? 10

Average Inventories at Destination How to calculate this when transit time varies? Inventory at Origin + Pipeline + Destination = Total Inventory 11

Maximum Inventory It is a random variable Bound it Calculate a distribution Tail Probabilities ... 12

Assignment #3 E(f(x))  f(E(x)) Avg. Cost of Shipment  Cost of Avg. Shipment Assignment 3: What relationship is there between Avg. Cost of Shipment Cost of Avg. Shipment? 13

LTL Rates Total Cost Volume Shipped Cost of Avg. Shipment Avg. Cost of 14 Customer 1 Volume Shipped Customer 2

Conclusion Typically the Cost of the Average Shipment Overestimates the Average Cost of a shipment. Calculating the Average Cost of a shipment is more tedious. Get Average Cost of Shipment from averages for each segment 15