Echelon Stock Formulation of Arborescent Distribution Systems: An Application to the Wagner-Whitin Problem

1
Echelon Stock Formulation of Arborescent
Distribution SystemsAn Application to
theWagner-Whitin Problem

• S. Armagan Tarim
• Department of Management, Hacettepe University,
  Ankara, Turkey.
• Ian Miguel
• AI Group, Department of Computer Science,
  University of York, UK.

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Distribution System Definition

• A supply chain of stocking points arranged in
  levels.
• Customer demands at level 1.
• Each level replenished from level above.
• Two costs Holding (c), procurement (c0).
• Supplier holding cost lt receiver holding cost.
• Given customer orders over some planning horizon
  of time periods.
• Find an optimal policy
• Set of decisions as to when and how much to
  order, minimising cost.

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Distribution Systems are Ubiquitous

• METRIC (Sherbrooke), MOD-METRIC (Muckstadt).
• Designed for the US Air Force.
• HP DeskJet Printer Supply Chain (Lee
  Billington).
• Optimizer (IBM).
• Global Supply-chain Model (DEC).

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Arborescent Distribution Systems

• Distribution system viewed as directed network.
• Nodes stocking points.
• Arcs flow of goods.
• We focus on arborescent (tree) structures
• Each node has at most one incoming link.
• Flows are acyclic.

F
Level i2
D
E
Level i1
A
B
C
Level i
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Wagner-Whitin Assumptions

• Demand
• Deterministic.
• Dynamic.
• Holding cost
• Linear in size of inventory.
• Ordering cost
• Constant (independent of order size).
• Holding, ordering costs fixed over planning
  horizon.
• Uncapacitated stocking points.
• 0 starting inventory, 0 delivery time.

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Modelling Conventional MIP Model

• Inventory (I), order quantity (X) variables
• One per stocking point, per time-period.
• Objective (T periods, N nodes)
• Minimise

Only incur procurement cost if order placed.
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Echelon Stock Formulation

• Echelon stocking point and all of its children.
• Echelon Stock (E) sum of stock in an echelon.

• Echelon holding cost (e)
• Incremental cost of holding stock at this node
  rather than its parent.

F
D
E
A
B
C
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Modelling Echelon MIP Model

• Inventory (E), order quantity (X) variables
• Objective (T periods, N nodes)
• Minimise

Demand (known) replaces order var.
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Echelon MIP Model Properties

• Previously known to be a valid model of serial
  distribution systems (Schwarz Schrage).
• Theorem Echelon MIP model valid for arborescent
  distribution systems.
• Gives a tighter relaxation than the conventional
  model.

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Adding Implied Constraints

• Conventional and echelon models can be improved
  by adding implied constraints.
• Follow logically from the initial model.
• But aid solver in pruning the search.
• IC1 In an optimal solution all stocking points
  must have 0 inventory at the end of the last
  period.
• Remaining stock incurs holding cost redundantly.

Conventional
Echelon
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Adding Implied Constraints

• IC2 In an optimal solution, if a parent node
  places an order, at least one of its children
  must also place an order.
• If no child makes an order, the parent node
  incurs a holding cost.
• Cost can be removed simply by delaying the order.

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Adding Implied Constraints

• IC3 Upper bound on conventional inventory
  variables (I, simple translation to E).
• Hold stock only if cheaper than ordering in next
  period.

Parent (m)
Parent (m)
stock
stock
Child (n)
Child (n)
stock
stock
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Adding Implied Constraints

• IC4 Upper bound for order variables (X) at the
  leaves of the distribution system.
• Order stock not absorbed by demand at current
  period only if cheaper than ordering later.
• Consider deferring for 1 period

• Demand varies over planning horizon
• Generalise to consider deferring an order into
  any of subsequent periods, finding minimum cost.
• Details in paper.

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Experiments

• Hypothesis
• Echelon model can yield improved results compared
  with conventional model.
• Test on different distribution structures.

Arborescent
Serial
Warehouse Retailer

• Details of test cases in paper.

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Results

• CPLEX8.1 Xpress 2003B
• Planning Horizon 10 to 18 periods

No proof of optimality in 30 problems out of
70 Allowed time 1 hour
Conventional MIP No ICs
All solved to optimality Max sol. Time 14.7
min On average 118 times faster
Conventional MIP ICs 1-2
Conventional MIP ICs 1-4
All solved to optimality Max sol. Time 2.7 min On
average 152 times faster
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Results
All solved to optimality Max sol. Time 11.5
min On average 45 times faster
Echelon MIP No ICs
All solved to optimality Max sol. Time 1.3 min On
average 753 times faster
Echelon MIP ICs 1-2
Echelon MIP ICs 1-4
All solved to optimality Max sol. Time 0.9 min On
average 951 times faster
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Results Summary

• Echelon model improves over conventional.
• IC1 (0 final inventory) and IC2 (parent only
  orders if one of children orders) give dramatic
  improvement.
• IC2 especially strong on serial systems.
• IC3 (inventory UB), IC4 (leaf order UB) also
  improve, but less dramatically.

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A Hybrid CP/LP Model

• Idea
• adds constraint propagation to reduce search
  further.
• Allows us to add further (non-linear) ICs.
• Models
• Conventional Echelon as shown before.
• With ICs1-4.
• Maintain for the LP
• Add the reification

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Adding Implied Constraints

• IC5 In an optimal solution, an order is only
  made at a stocking point whose inventory is 0.
• If order made at point t at a stocking point with
  some stock remaining, there was a holding cost
  from t-1 to t.
• Remove this cost simply by increasing order size
  at t.

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Adding Implied Constraints

• IC6 In an optimal solution, sizes of all orders
  composed from sums of demands of children
  (Zangwill).
• So, can enumerate the domains of the order (X)
  variables large reduction in domain size.
• Cost exponential in number of leaves beneath a
  node.
• So impractical in, for example, warehouse
  structure case.

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Results

• Ilog Hybrid 1.3 (SolverCplex).
• Hybrid takes longer than the MIP approaches.
• Time taken per node is 5 times that of MIP
  solvers.
• Search tree, however, often smaller than that
  generated by Xpress-MP (especially using IC6).
• Cplex uniformly better.

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Results

• Conventional vs. Echelon
• Advantage not as clear for hybrid.
• Largely positive, but sometimes echelon model
  gives worse performance.
• We know echelon gives tighter relaxation.
• Conjecture when results poor, due to
• Bad interaction with constraint propagation.
• Branching heuristic considers LP only.

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Conclusion

• Extended Schwarz Schrages (1978) proof of the
  validity of the echelon formulation for serial
  distribution systems to arborescent systems
• Confirmed the utility of this formulation in an
  MIP setting by empirical analysis using
  Wagner-Whitin problem
• Success of echelon formulation was less clear cut
  in conjunction with the hybrid CP/LP solver.
• Perhaps poor interaction with constraint
  propagation, and ill-informed heuristic.
• Under investigation!

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Resources

• Problem 40 at www.csplib.org.
• Entry includes
• Ilog Hybrid source code.
• Test instances.