Optimal Pricing and Inventory Policies in MultiEchelon Supply Chains Panos Seferlis
1
and
Lambros Pechlivanos 2
1Department
of Mechanical Engineering, Aristotle University of Thessaloniki 2Department of International and European Economic Studies Athens University of Economics and Business Athens, Greece
1st Southeast European Congress on Supply Chain Management
11-12 November 2011, Thessaloniki GR
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 multiechelon 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 Distribution centers Production sites Warehouse independent sites production lines
Retailer sites Order profile (prices) Satisfied demand
availability of resources
transportation costs between nodes inventory & storage costs transportation line capacity 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 Centralized Optimal Model-based Control System Distribution centers Production sites
Warehouse sites
Retailer sites Orders profile (prices) Satisfied demand
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 Balance in nodes without demand (intermediate) yi ,k t yi ,k t 1 xi ,k ,k t Lk ,k xi ,k ,k t k
k W , D, t T , i DP
k
Balance in nodes with demand (terminal) yi ,k t yi ,k t 1 xi ,k ,k t Lk ,k d i ,k t
k R, t T , i DP
k
Balance of unsatisfied orders BOi ,k t BOi ,k t 1 ri ,k t d i ,k t LOi ,k t
k R, t T , i DP
Selection of time period (hours, day, week) is based on system dynamics Approximation of discrete product units with continuous model A fraction of unsatisfied orders may be lost
Product demand ri ,k t rref ,i ,k rref ,i ,k
s ln ri ,k p j ,m t p ref , j ,m ri ,k t rref ,i ,k p ref , j ,m rref ,i ,k m j ln p j ,m R
DP
deterministic term
stochastic term
demand elasticities own elasticity (negative)
ln ri ,k i ,k ln pi ,k
cross product elasticity
positive: substitute negative: complementary
ln ri ,k i j ,k ln p j ,k
cross node own product elasticity (positive) ln ri ,k i , k m ln pi ,m
Demand elasticity own elasticity (negative) cross product elasticity
positive: substitute negative: complementary
ln ri ,k i ,k ln pi ,k
product price ↑ product demand ↓
ln ri ,k i j ,k ln p j ,k
product price ↑ sub product demand ↑ com product demand ↓
Demand elasticity cross node own product elasticity (positive)
•
ln ri ,k i , k m ln pi ,m
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 t t h
J t
kR i
t th
t
t th
t
t th
t
t Th
t
pi ,k t d i ,k t w BO BO ,i ,k
kR i
2 t i ,k
w
xi ,k,k t
w
yi ,k t
k W ,D ,R
k W ,D ,R
T ,i ,k ,k
Revenues generated from delivered products Cost of unsatisfied demand
Transportation costs
i
Y ,i ,k ,k
Inventory and storage costs
i
w
k W ,D ,R iDP
2 x t x t 1 x ,i ,k ,k i ,k ,k i ,k ,k
Suppression factor in product flows
Model predictive control principle
error, ek+2
desired trajectory model prediction embeds the effects of past and future (optimized) decisions rolling control horizon
tk-2
tk-1 tk tk+1 tk+2 tk+3
past control actions
uk-2
future control actions
uk-1 uk uk+1 uk+2 uk+3
Rolling horizon principle •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 tk-2
model prediction rolling rolling control control horizon ) ) horizon (t (tkk+1
tk-1 tk tk+1 tk+2 tk+3
past control actions
uk-2
desired trajectory
future control actions
uk-1 uk uk+1 uk+2 uk+3
• 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 Max p, x, y
J
s.t. Supply chain model Forecast equations Linearly constrained problem Bilinear terms (p * d) in performance index Lower bound on global solution by solving the nonconvex problem Upper 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 tree Adjiman, C. S., S. Dallwig, C. A. Floudas, and A. Neumaier, (1998) Comput. Chem. Eng., 22, 1137
Results – Case study Characteristics of network 2 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 Centralized Optimal Control System Distribution centers Production sites
Warehouse D1 sites Lag time 3 Lag time 4
S1
W1
W2
D2
Retailer sites
Lag time 2
R1
R2
Orders profile (prices) Satisfied demand
Network performance I Step change in demand for product A in R1 and R2 10
Delivered product / time period
14
Price per unit product
13 12 11 10 9
A(R1) B(R1) A(R2) B(R2)
8 7 0
20
40
60
Time periods
80
100
8
6
4
2
0 0
20
40
60
80
100
Time periods
A-B substitute products, no stochastic demand variation Prices for A initially increase whereas prices for B decrease to compensate for the increased A demand Gradually, the system reaches its final steady state
Network performance II
14
Substitute products
12
Substitute products
10
Complementary products
8 6
Complementary products
4
A(R1) B(R1) A(R2) B(R2)
2 0 0
Delivered products / time period
Price per product unit
Step change in demand for product A in R1 and R2
20
40
60
Time period
80
100
10
8
6
4
Complementary products
2
0 0
Substitute products 20
40
60
80
100
Time periods
Price decrease in B is deeper when A, B are complementary
Response to demand variations 4000
Price variation in each time period results in superior performance vs. constant product prices over entire horizon
Performance index
3800 3600 3400 3200
variable prices over horizon constant prices over horizon fixed prices
3000 2800 0
20
40
60
80
100
Time periods A-B substitute products deterministic step changes for A in R1 and R2 + stochastic demand variation
Price variability 12
Price per unit product
11.5
product B
11
constant prices over rolling horizon
10.5 10
product A
9.5 9 8.5 0
50
100
Time periods
150
200
Product price changes are smoother when compared to constant pricing over the rolling horizon
Effect of pricing policy I Performance index improvement (%)
18 Var 0.1 Var 0.2 Var 0.5
16 14 12 10 8 6 4 2 0
0
20
40
60
Time (periods)
80
100
The larger the stochastic variability in demand the greater the benefit when compared to a case without price manipulation
Effect of pricing policy II
Inventory standard deviation
6 Var 0.1 Var 0.2 Var 0.5
5
Node inventory variability is reduced significantly – Accommodated by proper pricing manipulation
4
3
2
1
0
0
2
4
6
8
10
Inventory nodes (#)
12
14
16
Effect of pricing policy III 1.3 Var 0.1 Var 0.2 Var 0.5 Var 1.0
Price standard deviation
1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3
1
2
3
4
5
6
7
8
Retailer nodes (#)
Mild variability for the product prices is observed
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