optimal control of one-warehouse multi-retailer systems with discrete demand m.k. doğru a.g. de kok...
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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)
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......
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]
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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]
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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
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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
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0
N
2
1
......
Analysis: Preliminaries
• Echelon stock concept• Echelon inventory position = Echelon stock + pipeline stock
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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
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.......... 2
Analysis: Echelon Costs
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Analysis: Costs attached to a period
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...
2
Analysis: Optimization Problem
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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.
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Myopic allocation
Balance Assumption: Allowing negative allocations
Analysis: Allocation Decision
• Example 1: N=3, identical retailers
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Balanced Allocation is feasible
Analysis: Allocation Decision
• Example 2: N=3, identical retailers
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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
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• 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: Allocation Decision
Analysis: Single Cycle Analysis
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Retailers: N=2
Analysis: Single Cycle Analysis
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Allocation Problem
– Necessary and sufficient optimality condition
– Incremental (Marginal) allocation algorithm
– is convex
Analysis: Single Cycle Analysis
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Warehouse
Optimal policy is echelon base stock policy
Infinite Horizon Problem
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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
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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
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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).
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......
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
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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
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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%
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Preliminary Results: Identical Retailers
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0 2 4 6 8 10 12 14
123456789101112131415
Sce
nar
ios
% Gaps : optimality gap
: (UB-LB)/LB*100
Analysis: Single Cycle Analysis
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Single-echelon:discrete
Two-echelon:continuous
Two-echelon:discrete