Echelon Stock Formulation of Arborescent Distribution Systems:

An 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.

• 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 < 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.

Distribution System: Definition

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).

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

D

A B C

E

Level i

Level i+1

Level i+2

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.

Modelling: Conventional MIP Model

• Inventory (I), order quantity (X) variables:• One per stocking point, per time-period.

• Objective (T periods, N nodes):• Minimise:

Tt Nn

ntnntn cIc 0

• Inventory constraint:

)(children'')1(:,

nntnnttnnt XXIINnTt

ntnt MXNnTt :,• Order placed?

Only incur procurement cost if order placed.

Echelon Stock Formulation

• Echelon: stocking point and all of its children.

• Echelon Stock (E): sum of stock in an echelon.•

F

D

A B C

E

)(children'':,

nntnntnt EIENnTt

• Echelon holding cost (e):• • Incremental cost of

holding stock at this node rather than its parent.

)(parent: nnn cceNn

Modelling: Echelon MIP Model

• Inventory (E), order quantity (X) variables:

• Objective (T periods, N nodes):• Minimise:

Tt Nn

ntnntn cEe 0

Demand (known)

replaces order var.

)(leaves'')1(:,

nntnnttnnt dXEENnTt

)(children'' 0:,

nntnnt EENnTt

• Inventory constraints:ntnt MXNnTt :,

• Order placed?

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.

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.

0: nTINn

0: nTENn

Conventional:

Echelon:

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.

ntnchildrenn

tnNnTt )('

':,

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.

stock

stock

Parent (m)

Child (n)

stock

stock

ntn Ic

Child (n)

Parent (m)

0nntm cIc

mn

nnt cc

cI

0:, NnTt

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.

)()(:leaves, 0ntntmnnntnt dXcccdXnTt

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.

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

Conventional MIPNo ICs

No proof of optimality in 30 problems out of 70Allowed time = 1 hour

Conventional MIPICs 1-2

All solved to optimalityMax sol. Time 14.7 minOn average 118 times faster

Conventional MIPICs 1-4

All solved to optimalityMax sol. Time 2.7 minOn average 152 times faster

Results

Echelon MIPNo ICs

Echelon MIPICs 1-2

All solved to optimalityMax sol. Time 1.3 minOn average 753 times faster

Echelon MIPICs 1-4

All solved to optimalityMax sol. Time 0.9 minOn average 951 times faster

All solved to optimalityMax sol. Time 11.5 minOn average 45 times faster

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.

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:

ntnt MXNnTt :,

)0(:, ntnt XNnTt

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.

)0()0(:},..2{ )1( tnnt IXNnTt

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.

Results

• Ilog Hybrid 1.3 (Solver+Cplex).

• 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.

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.

Conclusion

• Extended Schwarz & Schrage’s (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!

Resources

• Problem 40 at www.csplib.org.

• Entry includes:• Ilog Hybrid source code.• Test instances.