Optimal Control of One-Warehouse Multi-Retailer Systems with Discrete Demand M.K. Doğru A.G. de Kok G.J. van Houtum [email protected] [email protected].

Optimal Control of One-Warehouse Multi-Retailer Systems with

Discrete Demand

M.K. Doğru • A.G. de Kok • G.J. van Houtum

[email protected] • [email protected] • [email protected]

Department of Technology Management, Technische Universiteit Eindhoven

Eindhoven, Netherlands

System Under Study

• One warehouse serving N retailers, external supplier with ample stock, single item

• Retailers face stochastic, stationary demand of the customers• Backlogging, No lateral transshipments• Centralized control single decision maker, periodic review • Operational level decisions: when & how much to order

2

S 0

warehouse

C

C

C

....N

2

1

....

retailers

Literature

• Clark and Scarf [1960]– Allocation problem

– Decomposition is not possible, balance of retailer inventories

– Optimal inventory control requires solving a multi-dimensional

Markov decision process: Curse of dimensionality

– Solution is state dependent

• Eppen and Schrage [1981]– W/h cannot hold stock (cross-docking point)

– Base stock policy, optimization within the class

– Balance assumption (allocation assumption)

3

......

Literature

• Federgruen and Zipkin [1984a,b]– Balance assumption

– Optimality results for finite horizon problem, w/h is a cross-docking

point

– Optimality results for infinite horizon problem with identical retailers

and stock keeping w/h

• Diks and De Kok [1998]– Extension of optimality results to N-echelon distribution systems

• Literature on distribution systems is vast– Van Houtum, Inderfurth, and Zijm [1996]

– Axsäter [2003]

4

Literature

• Studies that use balance assumption:

Eppen and Schrage [1981], Federgruen and Zipkin

[1984a,b,c], Jönsson and Silver [1987], Jackson [1988],

Schwarz [1989], Erkip, Hausman and Nahmias [1990],

Chen and Zheng [1994], Kumar, Schwarz and Ward

[1995], Bollapragada, Akella and Srinivasan [1998],

Diks and De Kok [1998], Kumar and Jacobson [1998],

Cachon and Fisher [2000], Özer [2003]

5

Motivation

• Optimality results up to now are for continuous

demand distributions

This study aims to extend the results to

discrete demand distributions

• Why discrete demand?

– It is possible to handle positive probability mass at any point in

the demand distribution, particularly at zero.

– Intermittent (lumpy) demand

6

System Under Study

• W/h orders from an external supplier; retailers are replenished by shipments

• Fixed leadtimes• Added value concept• Backordering, penalty cost• Objective: Minimize

expected average holding and penalty costs in the long-run

7

0

N

2

1

......

Analysis: Preliminaries

• Echelon stock concept• Echelon inventory position = Echelon stock + pipeline stock

8

0

N

2

1

.....

Echelon inventory position of w/h

Echelon stock of w/h

Echelon stock of 2

Echelon inventory position of 2

..........

Analysis: Dynamics of the System

9

.......... 2

Analysis: Echelon Costs

10

Analysis: Costs attached to a period

11

...

2

Analysis: Optimization Problem

12

Analysis: Allocation Decision

• Suppose at the time of allocation ( t+l0 ), the sum of

the expected holding and penalty costs of the retailers

in the periods the allocated quantities reach their

destinations ( t+l0 +li ) is minimized.

13

Myopic allocation

Balance Assumption: Allowing negative allocations

Analysis: Allocation Decision

• Example 1: N=3, identical retailers

14

Balanced Allocation is feasible

Analysis: Allocation Decision

• Example 2: N=3, identical retailers

15

Balanced Allocation is infeasible

Analysis: Balance Assumption

• Interpretations

– Allowing negative allocations

– Permitting instant return to the warehouse without any

cost

– Lateral transshipments with no cost and certain leadtime

16

• Under the balance assumption, only depends on

the ordering and allocation decisions that start with an

order of the w/h in period t.

17

Analysis: Allocation Decision

Analysis: Single Cycle Analysis

18

Retailers: N=2

Analysis: Single Cycle Analysis

19

Allocation Problem

– Necessary and sufficient optimality condition

– Incremental (Marginal) allocation algorithm

– is convex

Analysis: Single Cycle Analysis

20

Warehouse

Optimal policy is echelon base stock policy

Infinite Horizon Problem

21

Newsboy Inequalities

• Existence of non-decreasing optimal allocation functions.

• Bounding

• Newsboy Inequalities– Optimal warehouse base stock level

– Newsboy inequalities are easy to explain to managers and non-mathematical oriented students

– Contribute to the understanding of optimal control

22

Conclusions

• Under the balance assumption, we extend the decomposition result and the optimality of base stock policies to two-echelon distribution systems facing discrete demands.– Retailers follow base stock policy– Shipments according to optimal allocation functions– Given the optimal allocation functions, w/h places orders following a base

stock policy

• Optimal base stock levels satisfy newsboy inequalities– Distribution systems with cont. demand: Diks and De Kok [1998]

• We develop an efficient algorithm for the computations of an optimal policy

23

Further Research

• N-stage Serial System with Fixed Batches

– Chen [2000]: optimality of (R,nQ) policies

– Based on results from Chen [1994] and Chen [1998] we

show that optimal reorder levels follow from newsboy

inequalities (equalities) when the underlying customer

demand distribution is discrete (continuous).

24

......

Further Research

• Eppen and Schrage [1981], Federgruen and Zipkin [1984a,b,c], Jönsson and Silver [1987], Jackson [1988], Schwarz [1989], Erkip, Hausman and Nahmias [1990], Chen and Zheng [1994], Kumar, Schwarz and Ward [1995], Bollapragada, Akella and Srinivasan [1998], Diks and De Kok [1998], Kumar and Jacobson [1998], Cachon and Fisher [2000], Özer [2003]

• Doğru, De Kok, and Van Houtum [2004] – Numerical results show that the balance assumption (that leads to the

decomposition; as a result, analytical expressions) can be a serious limitation.

• No study in the literature that shows the precise effect of the balance assumption on expected long-run costs

25

Further Research

• Optimal solution by stochastic dynamic programming– true optimality gap, precise effect of the balance assumption

– how good is the modified base stock policy

• Model assumptions– discrete demand distributed over a limited number of points

– finite support

• Developed a stochastic dynamic program

• Partial characterization of the optimal policy both under the discounted and average cost criteria in the infinite horizon– provides insight to the behavior of the optimal policy

– finite and compact state and action spaces

– value iteration algorithm

26

Preliminary Results: Identical Retailers

• Test Bed: 72 instances– N=2– w.l.o.g.– Parameter setting

– demand ~ [0,1,2,3,4,5]– LB-UB gap > 2.5%

27

Preliminary Results: Identical Retailers

28

0 2 4 6 8 10 12 14

123456789101112131415

Sce

nar

ios

% Gaps : optimality gap

: (UB-LB)/LB*100

Analysis: Single Cycle Analysis

29

Single-echelon:discrete

Two-echelon:continuous

Two-echelon:discrete