1st Southeast European Congress on Supply Chain Management 11-12 November 2011, Thessaloniki GR

Optimal Pricing and Inventory Policies in Multi- Echelon Supply Chains

Panos Seferlis 1

and

Lambros Pechlivanos 2

1Department of Mechanical Engineering, Aristotle University of Thessaloniki

2Department of International and European Economic StudiesAthens University of Economics and Business

Athens, Greece

Motivation -

Objectives

•

Congested transportation lines and heavy utilization of inventory nodes due to demand fluctuations may result in increased costs and lost orders

•

Product price manipulation can be used to alleviate congested routes and heavily utilized nodes by altering appropriately the demand profile at the end-points of the supply chain

•

We seek a method that will determine concurrently optimal pricing-inventory decisions for a multi-

echelon supply chain structure

•

Furthermore, we consider a multiproduct firm and hence we investigate the interactions among multiple substitute or complementary products

Supply chain management

• Objective: The satisfaction of the supply chain network goals through optimal inventory and pricing policies

• Method: The integration of the production, distribution and pricing problem for the entire network

• Innovative Features:

Product prices become an additional instrument for the fulfillment of the overall network objectives

Multi-echelon supply chain network

Production sitesindependent

production lines

Warehouse sites

Distribution centers

Retailer sites

Order profile (prices)Satisfieddemand

availability of resources

transportation costs between nodestransportation line capacity inventory & storage costs

inventory capacity

Optimal control strategy in supply chains

•

Optimal model-based control of the entire supply chain network

•

Difference equation model for supply chain behavior prediction

•

Forecast model for future stochastic variation of product demand

•

Control over a time horizon to ensure enhanced performance

•

Centralized scheme eliminates the propagation of flows variation to upstream nodes

•

Product price manipulation will absorb part of the demand variability

Integrated SC network optimization

Production sites

Warehouse sites

Distribution centers

Retailer sites

Orders profile (prices)

Satisfied demand

Centralized Optimal Model-based Control System

Modeling requirements

•

Dynamic network model (set of difference equations)

•

Selection of suitable time period (discretized time) based on the dynamics of the system

•

Definition of prediction time horizon•

Demand functions (elasticities)

•

Identification of stochastic model for product demand

•

Evaluation of inventory, storage and transportation cost factors

•

Evaluation of cost for unsatisfied demand

Supply chain network model

DPiTtDWktxLtxtytyk

kkikkk

kkikiki

,,,1 ,,,,,,,

DPiTtRktdLtx1tyty kik

kkkkikiki

,,,,,,,,

DPiTtRktLOtdtr1tBOtBO kikikikiki ,,,,,,,

Balance in nodes without demand (intermediate)

Balance in nodes with demand (terminal)

Balance of unsatisfied orders

Selection of time period (hours, day, week) is based on system dynamicsApproximation of discrete product units with continuous model

A fraction of unsatisfied orders may be lost

Product demand

k,i,ref

k,i,refsk,i

R

m

DP

j m,j,ref

m,j,refm,j

m,j

k,i

k,i,ref

k,i,refk,i

rrtr

pptp

plnrln

rrtr

demand elasticities

k,iplnrln

k,i

k,i

own elasticity(negative)

k,jiplnrln

k,j

k,i

cross product elasticity

positive: substitutenegative: complementary

mk,iplnrln

m,i

k,i

cross node - own product

elasticity(positive)

deterministic term stochastic term

Demand elasticity

k,iplnrln

k,i

k,i own elasticity

(negative)

k,jiplnrln

k,j

k,i cross product

elasticitypositive: substitute

negative: complementary

product price ↑product demand ↓

product price ↑sub product demand ↑com product demand ↓

Demand elasticity

mk,iplnrln

m,i

k,i

cross node - own product

elasticity(positive)

product price at any given node ↑

product demand at adjacent nodes ↑

•

Demand elasticity can be estimated through analysis of market data, survey results, direct experimentation by firm marketing department

Performance index

Cost of unsatisfied demand

Inventory and storage costs

Transportation costs

h

h

h

h

h

Tt

t R,D,Wk DPik,k,ik,k,ik,k,i,x

tt

t R,D,Wk ik,ik,k,i,Y

tt

t R,D,Wk ik,k,ik,k,i,T

tt

t Rk ik,ik,i,BO

tt

t Rk ik,ik,i

txtxw

tyw

txw

tBOw

tdtpJ

2

2

1

Revenues generatedfrom delivered products

Suppression factor in product

flows

Model predictive control principle

desired trajectory

model predictionembeds the effects of past and future (optimized) decisions

error, ek+2

tk+1 tk+2 tk+3tktk-1tk-2

rolling control horizon

uk+1 uk+2 uk+3ukuk-1uk-2

future control actionspast control actions

Rolling horizon principle

desired trajectory

model prediction

tk+1 tk+2 tk+3tktk-1tk-2

rolling control horizon (tk

)

uk+1 uk+2 uk+3ukuk-1uk-2

future control actionspast control actions

•At every time instance the first optimal decision is implemented•The rolling horizon shifts one period in the future•The size of horizon must allow dynamic effects of past actions to show in the trajectory rolling control

horizon (tk+1

)

• However, a very long horizon will make the system susceptible to stochastic variation of demand and other disturbances (e.g., failure of timely product transportation among nodes)

State/model update

• At the end of time period the actual demand (i.e., placed orders) is recorded and the inventory level at each level is updated

• The stochastic term in demand is updated using a forecasting model (ARIMA)

Solution of optimal control problem

equationsForecastmodelchainSupplys.t.

Max,,

Jyxp

Linearly constrained problemBilinear terms (p * d) in performance index Lower bound on global solution by solving the non-

convex problemUpper bound by solving a convex underestimation of the original problem Convergence achieved by successive subdivision of the region at each level in the branch and bound treeAdjiman, C. S., S. Dallwig, C. A. Floudas, and A. Neumaier, (1998) Comput. Chem. Eng., 22, 1137

Results –

Case study

Characteristics of network2 production sites (P) -

2 warehouse sites (W)

2

distribution centers

(D) -

4

retailer sites (R)

2 supplied products

transportation delay:

PW 4 periods, WD 3 periods, DR 2

periods

Problem size: 104 variables, 72

equations per time period

Scenarios

Step change for product A in R1 and R2

+ Stochastic variation in demand

IMA(1,1) forecast model

Integrated SC network optimization

Production sites

Warehouse sites

Distribution centers

Retailer sites

Orders profile (prices)

Satisfied demand

Centralized Optimal Control System

Lag time 4 Lag time 3

Lag time 2

W1

W2

D1

D2

R1

R2S1

0 20 40 60 80 1007

8

9

10

11

12

13

14

Time periods

Pric

e pe

r uni

t pro

duct

A(R1)B(R1)A(R2)B(R2)

0 20 40 60 80 1000

2

4

6

8

10

Time periodsD

eliv

ered

pro

duct

/ tim

e pe

riod

A-B substitute products, no stochastic demand variationPrices for A initially increase whereas prices for B decrease to

compensate for the increased A demandGradually, the system reaches its final steady state

Network performance IStep change in demand for product A in R1 and R2

Network performance II

0 20 40 60 80 1000

2

4

6

8

10

Time periodsD

eliv

ered

pro

duct

s / t

ime

perio

d0 20 40 60 80 100

0

2

4

6

8

10

12

14

Time period

Pric

e pe

r pro

duct

uni

t

A(R1)B(R1)A(R2)B(R2)

Price decrease in B is deeper when A, B are complementary

Step change in demand for product A in R1 and R2

Complementary productsComplementary products

Substitute products

Substitute products

Complementary products

Substitute products

0 20 40 60 80 1002800

3000

3200

3400

3600

3800

4000

Time periods

Perfo

rman

ce in

dex

Response to demand variations

variable prices over horizonconstant prices over horizonfixed prices

A-B substitute productsdeterministic step changes for A in R1 and R2

+ stochastic demand variation

Price variation in each time period

results in superior performance vs.constant product prices over entire

horizon

0 50 100 150 2008.5

9

9.5

10

10.5

11

11.5

12

Time periods

Pric

e pe

r uni

t pro

duct

constant prices over rolling horizon

product A

product B

Price variability

Product price changes are smoother when compared to constant pricing over the rolling horizon

Effect of pricing policy I

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

18

Time (periods)

Per

form

ance

inde

x im

prov

emen

t (%

) Var 0.1Var 0.2Var 0.5

The larger the stochastic variability in demand the greater the benefit when compared to a case without price

manipulation

Effect of pricing policy II

0 2 4 6 8 10 12 14 160

1

2

3

4

5

6

Inventory nodes (#)

Inve

ntor

y st

anda

rd d

evia

tion

Var 0.1Var 0.2Var 0.5

Node inventory variability is

reduced significantly –

Accommodated by proper

pricing manipulation

Effect of pricing policy III

Mild variability for the product prices is observed

1 2 3 4 5 6 7 80.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Retailer nodes (#)

Pric

e st

anda

rd d

evia

tion

Var 0.1Var 0.2Var 0.5Var 1.0

Concluding remarks

•

A framework for the solution of the integrated inventory and pricing policy problem for supply chain has been described

•

The proposed optimal control strategy for multi-echelon supply chain networks:•

calculates the optimal operating policy that maximizes revenues and customer service quality

•

calculates the optimal inventory policy•

calculates the optimal pricing policy

•

compensates effectively for stochastic demand variation•

compensates effectively for transportation delay

•

Manipulation of product prices absorbs portion of the demand variability and provides an additional instrument for the efficient management of supply chains