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?