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
vendor
retailer
products
sales & stocks
How often?
delivery cost
holding cost
frequently
large
small
less frequently
small
large
Deterministic demands over rounds
demands
Round
0
1
2
3
4
5
…
T
50
0
100
30
70
80
…
50
holding cost
100×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
In each round, we specify the route for visiting warehouses
Round 0
Round 1
…
Round T
•
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
non-decreasing
hv(t, u) ≤ hv(t’, u) for t’ ≤ t
•
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
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, …
•
Nested periodic schedule
Client A
Client B
Client C
Client D
every day
every week
every 2 weeks
every 4 weeks
fi+1 / fi ∈ Z
Partition v.s. Non-Partition
Round 2
Round 4
partition schedule
visit via the same route in each round
freq = 2
freq = 4
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)
Output
Input
•
undirected graph G = (V, E)
•
edge costs c: E → R≥0
•
root node s ∈ V •
penalties π: V − {s} → R≥0
rooted tree F minimizing
c(F) + π(V − V(F))
F
V− V(F)
Idea
holding costs
penelties
delivery costs
edge costs
IRP
PCST
IRP with nested policies → PCST
freq = f1
freq = f2
freq = f3
freq = fk
…
w=0
In the i-th copy:
T
w(ei ) = w(e) ·
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)
1
i
i+1
π(v, 1) + π(v, 2) + … + π(v, i) = H(v, i+1)
k
Monotone tree
A solution F for the PCST instance is monotone: vi ∈ F
monotone F
non-monotone F’
frequencies are nested
w(F) ≤ 2 w(F’)
vi+1∈ F
Outline of our algorithm
periodic schedule x
a monotone tree F
route cost of x = w(F)
holding cost of x = π(F)
Algorithm
1.
2.
3.
4.
Construct the PCST instance
Compute an approximate solution F to the PCST instance
Construct a monotone tree F’ from F
Output a schedule corresponding to F’
Theorem
Uncapacitated periodic IRP admits a 2ρ-approximation algorithm if the PCST problem admits a ρ-approximation algorithm.
Improve 2ρ to 2.55
PCST LP
min
s.t.
w> x
x( (Y )) + z(vi )
x, z 0
1,
8vi 2 8Y ✓ (
ST
⇤
V
)
\
{s
},
j=0 j
PCST LP + monotonicity constraints
min
s.t.
w> x
x( (Y )) + z(vi )
z(vi ) z(vi+1 ),
x, z 0
1,
ST
8vi 2 8Y ✓ ( j=0 Vj ) \ {s⇤ },
8v 2 V, 0  8i  T 1,
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
T
X
X
OPT
w(s, i)
di (t)/C
i2V
t=0
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?