The Integrated Warehouse-Inventory-Transportation Problem: A Stochastic Integer Quadratically-

Constrained Programming Approach

Christopher D. Hagmann , Nan Kong, Ph.D. Purdue University

Pratik J. Parikh, Ph.D. Data Analytics and Optimization Lab, Wright State University

CMMI #1235061 and #1235283

ICSP Bergamo XIII 12 July 2013

Introduction

Newest member of group (7 months)

First year PhD student

Studying Chemical Engineering at Purdue University

Newlywed

WAREHOUSE UTILIZATION

INVENTORY

TRANSPORTATIONM T W R F

OU

TB

OU

ND

INB

OU

ND

S1 S2 S3

W

V1 V2

S4

An illustration of the supply chain of an US-based apparel company

Common warehousing activities

Weekly variation in the units picked at the warehouse of the US-based apparel supply chain

Warehouse Weekly Workload Variation • Period: Jan – Dec 2011

• 42-219% variation in warehouse workload

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1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52

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Daily Workload Variation

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• A Fortune 500 Grocery Distributor

• Outbound activity at one of their warehouses in the US

• Period: Aug 29 – Sep 4, 2011

• Variation in workload: 76% - 153% of that week’s average

Integration of warehousing, inventory, and transportation decisions in a supply chain

The Integration of W, I, T

Inventory Transportation

Warehouse

Forward Impact

Reverse Impact

Impact of Technology used Workforce level

on Shipment schedules and quantity Inventory at warehouse and stores

Impact of Shipment schedules Shipment quantity Inventory levels

on Warehouse workload Workforce planning

Research Objectives

• Proactive vs. reactive decision making (warehouse perspective)

• Warehouse-Inventory-Transportation Problem (WITP)

Objective of WITP

Explore complex and dynamic interdependencies between warehouse, inventory, and transportation decisions

Determine the optimal distribution strategy while minimizing total cost

• Warehousing decisions: – Technology Selection (aisle configuration, layout, picking method, IT, etc.) – Workforce (permanent and temporary) – Other (cross-docking and cross-training)

The Math Formulation of WITP

WITP Formulation

WITP Formulation (cont’d)

Two-Stage Stochastic Integer

Quadratically-Constrained Programming (SIQCP) Extension

Incorporating Uncertainty

• Demand is an exogenous uncertainty.

• Full-time employees cannot be hired and fired with every time step.

• Technology is expensive and cannot be bought with every time step

• These motivate the need of a two-stage stochastic problem with full-time workforce level and technology usage decisions in the first-stage.

First-Stage Formulation

Second-Stage Formulation

Scenario-Wise Decomposition and Non-Anticipativity Constraints

In addition to all previously stated constraints

Dual Decomposition in Stochastic Integer Programming Carøe & Schultz

Instance Generation

• Randomly generate the mean for each demand

• Generate demand scenarios by sampling from a uniform distribution between 75% and 125% of the mean demand

• All demand variations are completely correlated and linked to previous time step

Conclusion

• Warehouses are a crucial part of supply chain logistics and should be included in overall optimization problems

• It is important to optimize warehousing decisions under uncertainty

Future Research

• Finish augmented Lagrangian relaxation code

• Investigate alternative scenario-wise decompositions

• Progressive Hedging

• Investigate alternative methods for handling quadratic constraints

• Investigate better methods for scenario generation

Thank You.