Optimal Control of One-Warehouse Multi-Retailer Systems with Discrete Demand

1
Optimal Control of One-Warehouse Multi-Retailer
Systems with Discrete Demand

• M.K. Dogru A.G. de Kok G.J. van Houtum
• m.k.dogru_at_tm.tue.nl a.g.d.kok_at_tm.tue.nl
  g.j.v.houtum_at_tm.tue.nl
• Department of Technology Management, Technische
  Universiteit Eindhoven
• Eindhoven, Netherlands

2
System Under Study
2

• 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

3
Literature
3

• 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)

4
Literature
4

• 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

5
Literature
5

• 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

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Motivation
6

• 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

7
System Under Study
7

• 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

1
0
2
......
N
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Analysis Preliminaries
8
1
Echelon inventory position of 2
Echelon stock of w/h
0
2
Echelon stock of 2
Echelon inventory position of w/h
.....
N

• Echelon stock concept
• Echelon inventory position Echelon stock
  pipeline stock

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Analysis Dynamics of the System
9
10
Analysis Echelon Costs
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Analysis Costs attached to a period
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12
Analysis Optimization Problem
12
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Analysis Allocation Decision
13

• Suppose at the time of allocation ( tl0 ), the
  sum of the expected holding and penalty costs of
  the retailers in the periods the allocated
  quantities reach their destinations ( tl0 li )
  is minimized.

Myopic allocation
Balance Assumption Allowing negative allocations
14
Analysis Allocation Decision
14

• Example 1 N3, identical retailers

Balanced Allocation is feasible
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Analysis Allocation Decision
15

• Example 2 N3, identical retailers

Balanced Allocation is infeasible
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Analysis Balance Assumption
16

• Interpretations
• Allowing negative allocations
• Permitting instant return to the warehouse
  without any cost
• Lateral transshipments with no cost and certain
  leadtime

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Analysis Allocation Decision
17

• Under the balance assumption, only
  depends on the ordering and allocation decisions
  that start with an order of the w/h in period t.

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Analysis Single Cycle Analysis
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Retailers N2
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Analysis Single Cycle Analysis
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Allocation Problem

• Necessary and sufficient optimality condition
• Incremental (Marginal) allocation algorithm
• is convex

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Analysis Single Cycle Analysis
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Warehouse
Optimal policy is echelon base stock policy
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Infinite Horizon Problem
21
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Newsboy Inequalities
22

• 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

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Conclusions
23

• 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

24
Further Research
24

• 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).

25
Further Research
25

• 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
• Dogru, 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

26
Further Research
26

• 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

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Preliminary Results Identical Retailers
27

• Test Bed 72 instances
• N2
• w.l.o.g.
• Parameter setting
• demand 0,1,2,3,4,5
• LB-UB gap gt 2.5

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Preliminary Results Identical Retailers
28
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Analysis Single Cycle Analysis
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Two-echelon discrete
Two-echelon continuous
Single-echelon discrete