Statics and Dynamics of Complex Network Systems: Supply Chain Analysis and Financial Networks with

Intermediation

Ke Ke

MKIDS Mini-WorkshopSeptember 10, 2003

The Virtual Center for Supernetworks

Introduction

• Literature Survey• Related Publications• Research Motivation

Literature Survey

Supply Chain Analysis:

• Nagurney and Dong (2002)

• Nagurney, Dong, and Zhang (2001)

• Nagurney, Loo, Dong, and Zhang (2001)

• Nagurney and Dong (2000)

• Nagurney, Dong, and Mokhatarian (2000, 2001)

Financial Networks with Intermediation:

• Thore (1969, 1980)• Nagurney, Dong, and

Hughes (1992)• Nagurney and Dong

(1995, 1996a, b)

Related Publications

• Nagurney, Ke, Cruz, Hancock, and Southworth (2002)

• Nagurney and Ke (2001a, b)• Nagurney and Ke (2003)

Research Motivation• We propose a multilevel network to capture

distinct flows, in particular, the logistical, informational, and financial flows within the same supply chain network system.

• The theoretical framework provides an abstraction of decision-making in the case of agents, such as firms, consumers, as well as intermediaries, who operate in a decentralized fashion on both physical and virtual networks, and allows not only for the visualization of business processes as networks but also for the prediction of production volumes and shipments of products, prices, as well as, as appropriate information and financial flows.

Research Motivation (Cont.)

• We develop a framework for the modeling, analysis, and computation of solutions to financial network problems in the presence of intermediation.

• Advances in telecommunications have had an enormous effect on financial services and the options available for financial transactions. We then will further to include the electronic transaction in the financial network modeling.

Dynamics of Supply Chains: a Multilevel Network

Prospective

Notations

• m firms with a typical firm denoted by i,• n retailers with a typical retailer

denoted by j,• o demand markets with a typical

demand market denoted by k.

The Demand Market Price Dynamics

The Dynamics of the Commodity Shipments

Between the Retailers and the Demand Markets

The Dynamics of the Prices at the Retail Outlets

The Dynamics of Commodity Shipments BetweenFirms and Retailers

The Projected Dynamical System

Let X = the shipment and price variables

and F(X) (Fij, Fjk, Fj, Fk) i=1,…,m; j=1,…,n; k=1,…,o,

Where ; jij

j

ij

ijij

ij

iij q

Qc

q

qc

q

QfF 2

11 )()()(

F

;)( 32

2 kjkjjk QcF

).(, 3111

k

n

jjkk

o

kjk

m

iijj dqFandqqF

Then the dynamic model can be rewritten as the PDS defined by the initial value problem:

X = ΠK(X, -F(X)), X(0)=X0.

The Equilibrium Point

Qualitative Properties

• Monotonicity• Lipschitz continuity• Existence and uniqueness of a solution

to the initial value problem• Stability of the system

A Discrete Time Algorithm

• Step 0: Initialization

• Step 1: Computation

• Step 2: Convergence verification

Step 1: Computation

Compute (Q1, Q2, , ) by solving the variational inequality:

onnomnR

23

Financial Networks with Intermediation

Introduction

Financial network with intermediation

• consists of:– source of funds– intermediaries– consumers of

funds• the flows on the

links: the allocation of the financial instruments

Multilevel supply chain network

• consists of:– manufacturers – retailers – consumer markets

• the flows on the links: prices of products

Introduction (Cont.)

• direction of financial flows: from top tier, through middle tier, until the bottom tier.

• objective of the decision-makers of first two tiers– profit maximization– minimization of

risks

• direction of financial flows: – bottom up

• objective of the decision-makers of first two tiers:– Profit maximization

The Behavior of Source Agents

The Optimality Conditions for Source Agents

The Behavior of Financial Intermediaries

The Optimality Conditions for All Intermediaries

The Behavior of Consumers at Demand Markets

The Optimality Conditions for

Demand Markets

The Equilibrium Conditions

Qualitative Properties

• Monotonicity• Existence of a solution to the VI problem• Uniqueness

• Lipschitz continuity

The Algorithm

Financial Networks with Electronic Transactions

The Network Structure

• electronic transactions • source agents can transact:

– with the intermediaries either physically or electronically

– with the consumers in an electronic manner.

• intermediaries transact with the consumers either physically or electronically

Notations

• l: the transaction mode– l=1: a physical transaction – l=2: an electronic transaction via the

Internet

• All other notations and assumptions remain the same as in the previous model.

The Behavior of Source Agents

The Optimality Conditions for Source Agents

The Behavior of Financial Intermediaries

The Optimality Conditions for

All Intermediaries

The Behavior of Consumers at Demand Markets

The Optimality Conditions for

Demand Markets

The Equilibrium Conditions

Qualitative Properties

• Monotonicity• Existence of a solution to the VI problem• Uniqueness

• Lipschitz continuity

The Algorithm

• Step 0: Initialization

• Step 1: Computation

• Step 2: Convergence verification

Step1: Computation

Numerical Examples

Numerical Examples (Cont.)

Numerical Examples (Cont.)

Summary• The dynamic supply chain network model and a

series of financial equilibrium models are developed here. Sufficient progress has already been obtained as evidenced from our publications.

• We extend the work in general financial equilibrium modeling, analysis, and computation in a network framework to include financial intermediaries, the underlying dynamics associated with the financial flows and prices, and the inclusion of electronic transactions. It, thus, will incorporate the dimension, lacking in financial flow of funds accounts, of the explicit dynamic behavior of the various financial agents as well as the price dynamics.

Thank You!

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http://supernet.som.umass.edu/