Columbia Express

Algorithms and heuristics used by shipping company

Yoon Keun AneJeongwho Lee

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Service Description

• 24/7 shipping company with distribution centers across the U.S.

• Provides 24-hour service, 2-days shipping services

• If late, the company refunds shipping fees to the customers

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Company Objective & Constraints

• Objective : minimize overall costs including shipping costs and penalty costs

• Constraints– Deadlines– Carrier Capacity

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How to approach the problem

3 stages of the problem1.Package Center to Main Dist. Center

- Package Shipment Schedule (Knapsack)2.MDC to Local Dist. Centers

- Minimizing Carrier Cost (Bin-Packing)3.LDC to Final Destination

- Minimizing Delivery Cost (TSP)

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Co-ExpressProblem Delineation

• One MDC• Many customers• Local distribution

centers at each zone (zones divided by customer population)

• Optimization done by each zone

I. Shipment Schedule to MDC

1. Package received at Columbia2. Package scheduled by deadline to be send to MDC

Use the following integer programming to maximize the total weight of jobs that finish before their deadlines

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Integer Programming

• Cj = shipping cost based on weight, distance and deadline

• Wj= Physical weight of the package• This problem is NP-complete problem also called

knapsack problem, and can be solved by dynamic programming

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max ∁jXj s.t PjXj ≤ dj ∶ deadline WjXj ≤ Cj ∶ Capacity j

Xj ∈ሼ0,1ሽ

II. MDC to LDC

• Purpose : Minimize number of trucks according to its size

• Motivation : No need to use huge trucks nor many trucks going from MDC to LDC

• Solution: “Bin-packing” Problem : Maximize utilization space in each container

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Graphical Interpretation

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Figure 1 Figure 2

• We need to minimize the size of empty space which is shown by white spaces in figure 2

Column Generation Method

• CGM is used to solve bin-packing• Dual variables from original LP is used to

create new column of constraints that will give a better solution than orig. LP

• Repetition of above steps will lead to finding the optimal pattern

• Formulation used most frequently nowadays

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Bin Packing Approx.

• First Fit Label bins as 1, 2, 3, . . .Objects are considered for packing in the order 1, 2, 3, . . .Pack object i in bin j where j is the least index such that bin j can contain object i.

• Best Fit The same as FF, except that when object i is to be packed, find the bin which after accommodating object i will have the least amount of space left.

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III. Local Delivery

• Purpose: Minimize the total distance traveled by local delivery trucks

• Well-known NP-hard Traveling Salesman Problem

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Solving TSP

• TSP can be solved using various methodsex) Algorithm 4.4.4.5

TSP relaxation Dynamic Programming

• We adopted TSP relaxation method to arrive at near-optimal solution

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TSP relaxation

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Limitations & Future Research

• Stochastic demand and processing time

• Hard deadline was not considered when solving II & III

• Minimizing cost at distribution intsct• Application to real-life data

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THANK YOU, ANY QUESTIONS?

THANK YOU, ANY QUESTIONS?