# Reoptimization framework and policy analysis for maritime inventory routing under uncertainty

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## Abstract

We study a maritime inventory routing problem, in which shipments between production and consumption nodes are carried out by a fleet of vessels. The vessels have specific capacities and can be chartered under different agreements. The inventory levels of all consumption nodes and some production nodes should be maintained within specified bounds; for the remaining production nodes, orders should be picked up within pre-defined time windows. We propose a discrete-time mixed-integer programming model. In the face of new information and uncertainty, this optimization model has to be re-solved, as the horizon is rolled forward. We discuss how to account for different sources of uncertainty. We present a rolling-horizon reoptimization framework that allows us to study different policies that impact the quality of the implemented solution, so we can identify the optimal set of policies.

## Keywords

Mixed-integer programming Optimization under uncertainty Inventory routing Rolling horizon approach## List of symbols

## Indices/sets

*d*∈**D**Dates (absolute time)

*i*∈**I**Time-chartered vessels

*j*∈**J**Nodes in the SC network, including vessel center (

*vc*)- (
*j, j′*)∈**A**\(\subseteq {\mathbf{J}} \times {\mathbf{J}}\) Arcs (trips) in the SC network

- \(k \in {\mathbf{K}}_{j}\)
Orders of third-party production node

*j**l*∈**L**Node clusters

*m*∈**M**Products

*n*∈**N**Number of unreserved vessels/trips

*t*∈**T**Time points or periods

## Subsets

- \({\mathbf{A}}_{l}\)
Arcs (trips) that are within cluster

*l*- \({\mathbf{A}}_{t}^{R}\)
Arcs (trips) already reserved to be served using voyage charters at time point

*t*- \({\mathbf{I}}^{R}\)
Time charters located at the vessel center but reserved

- \({\mathbf{J}}^{P} /{\mathbf{J}}^{C}\)
Production/consumption nodes

- \({\mathbf{J}}^{TP} /{\mathbf{J}}^{OP}\)
Third-party/owned production nodes

## Binary variables

- \(W_{{ijj^{'} t}}^{L}\)
=1 if time-chartered vessel \(i\) starts a trip from

*j*to*j′*at time point*t*- \(W_{{jj^{'} t}}^{S}\)
=1 if a voyage-chartered vessel starts a trip from

*j*to*j′*at time point*t*- \(\bar{X}_{ijt}^{L}\)
=1 if time charter

*i*is at node*j*during time period*t*- \(\bar{Y}_{it}^{L}\)
=1 if vessel

*i*is chartered in period*t*beyond \(\vartheta^{L}\) periods

## Non-negative variables

- \(C^{ALL} /C_{t}^{OF} /C_{t}^{UF}\)
Total/overflow/underflow cost

- \(C_{t}^{MH} /C_{t}^{FT} /C_{t}^{VT}\)
Material holding/fixed transportation/variable transportation cost

- \(C_{t}^{FL} /C_{t}^{EL} /C_{t}^{S}\)
Fixed time-charter/extended time-charter/voyage-charter cost

- \(C_{t}^{EP}\)
Penalty term for modeling early pick-up preference

- \(F_{{ijj^{'} mt}}^{L}\)
Product

*m*in time-chartered vessel*i*traveling from*j*to*j′*starting at time point*t*- \(F_{{jj^{'} mt}}^{S}\)
Product

*m*in the voyage-chartered vessel from*j*to*j′*starting at time point*t*- \(L_{jmt}\)
Inventory level of product

*m*at node*j*at time point*t*- \(L_{jmt}^{OF} /L_{jmt}^{UF}\)
Overflow/underflow amount of product

*m*of node*j*at time point*t*

## Parameters

- \(\alpha\)
Confidence level

- \(\beta\)
Penalty constant for modeling early pick-up preference

- \(\gamma_{i}^{MAX} /\gamma^{MAX}\)
Capacity of time-chartered vessel

*i*/voyage-chartered vessels- \(\gamma_{i}^{MIN} /\gamma^{MIN}\)
Minimum load on time charter

*i*/voyage charter when traveling from a production node to a consumption node- \(\delta\)
Time period length

- \(\delta^{LE}\)
Earliest time a time charter becomes available

- \(\delta_{n}^{L} /\delta_{ln}^{S}\)
Time when the

*n*th time/voyage charter becomes available- \(\varepsilon_{L}\)
Probability of time charter availability

- \(\zeta_{jmt}^{MAX} /\zeta_{jmt}^{MIN}\)
Maximum/minimum level of product

*m*at node*j*at time point*t*- \(\eta\)
Planning horizon

- \(\theta_{jkt}\)
=1 if period

*t*is in pick-up window*k*of third-party production node*j*- \(\vartheta^{L}\)
Minimum duration of time charter rental

- \(\lambda^{LA} /\lambda^{LB} /\lambda^{LR}\)
Earliest reservation/latest reservation/returning notice time for time charters

- \(\lambda^{SA} /\lambda^{SB}\)
Earliest/latest reservation time for voyage charters

- \(\lambda^{PU}\)
Time when a pick-up window becomes deterministically known

- \(\xi_{{ijj^{'} }}^{MAX} /\xi_{{jj^{'} }}^{MAX}\)
Maximum allowable load for trip (

*j,j′*) using time charter*i*/voyage charter- \(\pi_{jm}^{MH} /\pi_{jmt}^{OF} /\pi_{jmt}^{UF}\)
Material holding/overflow/underflow cost

- \(\pi_{{jj^{'} }}^{FT} /\pi_{{jj^{'} }}^{VT}\)
Fixed/variable transportation cost for trip (

*j,j′*)- \(\pi_{i}^{FL} /\pi_{i}^{EL}\)
Standard/extension cost for time charters

- \(\pi_{jt}^{EP}\)
Penalty used to model early pick-up preference

- \(\pi_{{jj^{'} }}^{S}\)
Voyage charter cost for trip (

*j,j′*)- \(\rho_{jmt}\)
Production (positive) or consumption (negative) rate of node

*j*during period*t*- \(\sigma_{jk}^{OS} /\sigma_{jk}^{OE}\)
Start/end time of the pick-up window of order

*k*from third-party production node*j*- \(\sigma^{SW}\)
Soft window length

- \(\tau_{{jj^{'} }}\)
Traversal time along arc (

*j*,*j′*)- \(\varphi_{jmk}\)
Amount of product

*m*in order*k*from third-party production node*j*- \(\chi_{it}\)
=1 if period

*t*is within the first \(\vartheta^{L}\) periods of the current time charter of vessel*i*- \(C^{ID} \left( d \right)\)
Estimated cost at date

*d*- \(\hat{F}_{ijmt}^{L}\)
Amount of product

*m*in vessel*i*en route to node*j*, expected to arrive at time point*t*- \(\hat{F}_{{j^{'} jmt}}^{S}\)
Amount of product

*m*that is en route and will arrive at node*j*at time point*t*from production node*j′*from the voyage charter- \(\hat{W}_{{ijj^{'} t}}^{L}\)
=1 if vessel

*i*is scheduled to depart*j*towards*j′*at time point*t*- \(\hat{W}_{{jj^{'} t}}^{S}\)
=1 if a voyage charter is scheduled to depart

*j*for*j′*at time point*t*- \(\hat{X}_{ijt}^{L}\)
=1 if vessel

*i*is at node*j*initially (*t*= 0), or it is en route and will arrive at*j*at time point*t*(*t*> 0)

## Notes

### Acknowledgements

The authors would like to acknowledge financial support from the US National Science Foundation under grant CBET- 1264096.

## Supplementary material

## References

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