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Deliver or Hold: Approximation Algorithms for the Periodic Inventory Routing Problem
Takuro Fukunaga (National Institute of Informatics)
!
joint work with
Afshin Nikzad (Stanford University) R. Ravi (Carnegie Mellon University)
Vendor managed inventory (VMI) model
retailervendor
How often?
delivery cost holding cost
frequently large small
less frequently small large
products
sales & stocks
Deterministic demands over rounds
Round 0 1 2 3 4 5 … T
50 0 100 30 70 80 … 50
demands
holding cost100×h(0, 2) 70×h(3, 4) + 80×h(3, 5)
h(i, j): cost for holding a single unit of products in rounds i through j
Routing in each round
Round 1Round 0 Round T
In each round, we specify the route for visiting warehouses
• WLOG, route in a single round is a set of trees rooted at
• capacitated setting: total delivery in each tree ≤ vehicle capacity
…
Inventory Routing Problem (IRP)
Input
• metric (V, w)
• depot s ∈ V
• holding cost hv(t, t’) for v ∈ V, t, t’ ∈ {0, …, T}
• demand dv(t) for v ∈ V, t ∈ {0, …, T}
• vehicle capacity C
Output
• a set of trees rooted at s in each round
• allocation of demands to trees
non-decreasing
hv(t, u) ≤ hv(t’, u) for t’ ≤ t
a demand dv(t) cannot be divided
Inventory Routing Problem (IRP)
Constraints
• demand constraint:
each demand is allocated to a tree in the same or earlier rounds
• capacity constraint: each tree is allocated ≤C units of demands
Open: Is there a constant approximation algorithm?
Known:
• polylog(|V|)-approximation
• constant approximations for Joint Replenishment Problem
= two level trees, e.g., [Levi et al. 2008]
Our results: constant-approximation for periodic schedules
Periodic schedule
(General) Periodic schedule• Every vertex has the same demand in all rounds (i.e. dv(t) = dv(t’))
• Available frequencies f1, …, fk are given
• A solution allocates a frequency fi to each vertex, and visits it
in rounds 0, fi, 2fi, …
Client A
Client B
Client C
Client D
every day
every week
every 2 weeks
every 4 weeks
Nested periodic schedule
fi+1 / fi ∈ Z
Partition v.s. Non-PartitionRound 2
freq = 2
freq = 4
Round 4
partition schedule
visit via
the same route in each round
non-partition schedule
Our results
Uncapacitated schedules
• 2.55-approx algorithm for uncapacitated nested periodic schedules
• 4-approx algorithm for uncapacitated nested partition schedules
• 8-approx algorithm for uncapacitated partition schedules
Capacicated schedules
γ-approx for uncapacitated schedules ⇒ (γ + 2)-approx for capacitated schedules
Structural results
relationships between various schedules
Prize-collecting Steiner tree (PCST)
Input
• undirected graph G = (V, E)
• edge costs c: E → R≥0
• root node s ∈ V
• penalties π: V − {s} → R≥0
Output
rooted tree F minimizing
c(F) + π(V − V(F))
V− V(F)
F
Idea
IRP PCST
edge costs
holding costs penelties
delivery costs
IRP with nested policies → PCST
…
freq = f1 freq = f2 freq = f3 freq = fk
In the i-th copy:
w = 0
w(ei) = w(e) ·T
fi
Setting of penalties
• H(v, i): holding cost when v is assigned frequency fi
π(v, 1) := H(v, 1)
π(v, i) := H(v, i+1) − H(v, i)
i i+1
π(v, 1) + π(v, 2) + … + π(v, i) = H(v, i+1)
1 k
Monotone tree
A solution F for the PCST instance is monotone: vi ∈ F ⇒ vi+1∈ F
monotone F
non-monotone F’
frequencies are nested ⇒ w(F) ≤ 2 w(F’)
Outline of our algorithm
Algorithm
1. Construct the PCST instance
2. Compute an approximate solution F to the PCST instance
3. Construct a monotone tree F’ from F
4. Output a schedule corresponding to F’
TheoremUncapacitated periodic IRP admits a 2ρ-approximation algorithm if the PCST problem admits a ρ-approximation algorithm.
periodic schedule x ⇔ a monotone tree Froute cost of x = w(F)
holding cost of x = π(F)
Improve 2ρ to 2.55
PCST LP
min w>x
s.t. x(�(Y )) + z(vi) � 1, 8vi 2 8Y ✓ (ST
j=0 Vj) \ {s⇤},z(vi) � z(vi+1), 8v 2 V, 0 8i T � 1,x, z � 0
min w>x
s.t. x(�(Y )) + z(vi) � 1, 8vi 2 8Y ✓ (ST
j=0 Vj) \ {s⇤},x, z � 0
PCST LP + monotonicity constraints
Theorem
Threshold rounding gives a 2.55-approximate monotone tree.
Capacitated IRP
1. Solve uncapacitated IRP 2. Divide each tree into subtrees
3. Connect a tree to the root by augmenting a shortest path
OPT �X
i2Vw(s, i)
TX
t=0
di(t)/C
⇒ shortest paths ≤ 2OPT
Conclusion
Our contributions
• IRP: New optimization problem that combines routing and inventory management problems
• Several constant approximation algorithms for periodic schedules
Open problems
• Constant approximation for general case?