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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.

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Abbreviations

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

\({\mathbf{A}}_{l}\) :

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

\(W_{{ijj^{'} t}}^{L}\) :

=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

\(C^{ALL} /C_{t}^{OF} /C_{t}^{UF}\) :

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

\(\alpha\) :

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 nth 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)

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Chemical and Biological Engineering, University of Wisconsin – Madison, 1415 Engineering Drive, Madison, WI, 53706, USA

    Yachao Dong & Christos T. Maravelias

  2. BP plc, 150 West Warrenville Road, Naperville, IL, 60563, USA

    Norman F. Jerome

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  1. Yachao Dong
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  2. Christos T. Maravelias
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  3. Norman F. Jerome
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Correspondence to Christos T. Maravelias.

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Appendix: Algorithms

Appendix: Algorithms

Algorithm A1 Check time charter availability

If all desired time-chartered vessels are available, the algorithm ends with resolve equal to no; otherwise, resolve returns yes. In line 5, \(\varepsilon_{L} (t)\) denotes the probability that a vessel is available at time \(t\). In line 10, \(\lambda^{LC}\) denotes the earliest time a time charter is guaranteed to be available. In the case study, \(t_{LA} = 7,t_{LB} = 21,\varepsilon_{L} (t_{LA} ) = 0.85, \varepsilon_{L} (t_{LB} ) = 1,\lambda^{LC} = 21\).

figure b

Algorithm A2 Update availability of voyage charters

The availability profiles of voyage charters are cluster-specific. For each cluster l, we (1) remove the vessels that were reserved in the previous period (lines 2–3); (2) update the times when vessels become available (lines 4–15); (3) sort those times in ascending order (line 16); and (4) generate new availability profile (lines 17–19). In line 2, \(nlast_{l}\) is the number of trips in cluster l reserved in the last period. Parameters \(\varepsilon_{SA} , \varepsilon_{SB} , \varepsilon_{SC} , \varepsilon_{SD}\) denote the probability that the time a vessel becomes available remains unchanged, decreases by 1 period, increases by 1 period, and increases by 2 periods, respectively. For example, in the case that the time a vessel becomes available is unchanged, the new \(\delta_{ln}^{S}\) is the old \(\delta_{ln}^{S}\) minus one (line 7), because the horizon has been rolled forward by one day. It is also possible that a previously available vessel is reserved by another party, and thus becomes unavailable, as shown in line 15. In the case study, \(\varepsilon_{SA} = 0.75,\varepsilon_{SB} = 0.05,\varepsilon_{SC} = 0.05,\varepsilon_{SD} = 0.05, newA = 9,newB = 12\).

figure c

Algorithm A3 Check availability of voyage charters

The availability of voyage charters is checked for each cluster; if some desired vessels are not available, resolve returns yes.

figure d

Algorithm A4 Update parameters due to trip delays

There are three types of trip delays: (1) a reserved vessel that is yet to arrive can be late for 1 period with a probability of \(\varepsilon_{DI1}\) or 2 periods with a probability \(\varepsilon_{DI2}\) (lines 1–6 and 7–12 respectively for time and voyage charters); (2) an ongoing trip can have a delay of 1/2 periods with a probability \(\varepsilon_{DO1} /\varepsilon_{DO2}\) (lines 13–18 and 19–24 for the time- and voyage-chartered vessels, respectively); and (3) a pick-up/delivery can be 1/2 period longer with a probability \(\varepsilon_{DT1} /\varepsilon_{DT2}\) (lines 25–32,33–40 respectively for time and voyage charters). Note that even though the probability of delay at each time period does not depend on trip length, longer trips tend to have larger delays, because they have more time periods during which delays can be observed. In the case study, \(\varepsilon_{DI1} = 0.15,\varepsilon_{DI2} = 0.05,\varepsilon_{DO1} = 0.08,\varepsilon_{DO2} = 0.02,\varepsilon_{DT1} = 0.15,\varepsilon_{DT2} = 0.05.\)

figure e

Algorithm A5 Update pick-up windows

For pick-up windows whose estimated start time is \(t = \lambda^{PU}\), the actual start time and window length are specified. Parameter newC is the maximum adjustment of the start time; newD/newE is the shortest/longest window length. In the case study, \(newC = 3,newD = 2,newE = 3\).

figure f

Algorithm A6 Update initial inventory

The real-time initial inventory level is updated using a normal distribution. The consumption/production rate in the last period is included in \(L_{jm1}\). In the case study, \(\sigma_{AF} = 0.05\).

figure g

Algorithm A7 Update forecast consumption/production rate

Consumption/production forecast rate are updated using a normal distribution. In the case study, \(\sigma_{FF} = 0.05\).

figure h

Algorithm A8 Fix variables according to decisions made previously

Variables related to three types of decisions are fixed. First, the vessel company should be notified \(\lambda^{LR}\) periods before returning a time charter (lines 1–2). Second, decisions regarding time charters made previously are fixed (lines 3–4); \(\delta^{LE}\) is the earliest time when a time charter becomes available (updated in lines 20–23 in Algorithm 1). Finally, decisions regarding voyage charters are fixed (lines 5–8). In the case study, \(\lambda^{LR} = 15\).

figure i

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Dong, Y., Maravelias, C.T. & Jerome, N.F. Reoptimization framework and policy analysis for maritime inventory routing under uncertainty. Optim Eng 19, 937–976 (2018). https://doi.org/10.1007/s11081-018-9383-8

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