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

- 39 Downloads

## 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

- Agra A, Andersson H, Christiansen M, Wolsey L (2013) A maritime inventory routing problem: discrete time formulations and valid inequalities. Networks 62(4):297–314MathSciNetCrossRefMATHGoogle Scholar
- Agra A, Christiansen M, Delgado A, Hvattum LM (2015) A maritime inventory routing problem with stochastic sailing and port times. Comput Oper Res 61:18–30MathSciNetCrossRefMATHGoogle Scholar
- Agra A, Christiansen M, Delgado A (2017) Discrete time and continuous time formulations for a short sea inventory routing problem. Optim Eng 18(1):269–297MathSciNetCrossRefMATHGoogle Scholar
- Al-Ameri TA, Shah N, Papageorgiou LG (2008) Optimization of vendor-managed inventory systems in a rolling horizon framework. Comput Ind Eng 54(4):1019–1047CrossRefGoogle Scholar
- Andersson H, Hoff A, Christiansen M, Hasle G, Løkketangen A (2010) Industrial aspects and literature survey: combined inventory management and routing. Comput Oper Res 37(9):1515–1536MathSciNetCrossRefMATHGoogle Scholar
- Balasubramanian J, Grossmann IE (2004) Approximation to multistage stochastic optimization in multiperiod batch plant scheduling under demand uncertainty. Ind Eng Chem Res 43(14):3695–3713CrossRefGoogle Scholar
- Bassett MH, Pekny JF, Reklaitis GV (1997) Using detailed scheduling to obtain realistic operating policies for a batch processing facility. Ind Eng Chem Res 36(5):1717–1726CrossRefGoogle Scholar
- Braun MW, Rivera DE, Flores ME, Carlyle WM, Kempf KG (2003) A model predictive control framework for robust management of multi-product, multi-echelon demand networks. Annu Rev Control 27(2):229–245CrossRefGoogle Scholar
- Campbell AM, Savelsbergh MWP (2004) A decomposition approach for the inventory-routing problem. Transp Sci 38(4):488–502CrossRefGoogle Scholar
- Christiansen M, Fagerholt K (2002) Robust ship scheduling with multiple time windows. Naval Res Logist (NRL) 49(6):611–625MathSciNetCrossRefMATHGoogle Scholar
- Christiansen M, Nygreen B (2005) Robust inventory ship routing by column generation. In: Desaulniers G, Desrosiers J, Solomon MM (eds) Column generation, Springer, Boston, MA, pp 197–224. https://doi.org/10.1007/0-387-25486-2_7 CrossRefGoogle Scholar
- Coelho LC, Cordeau JF, Laporte G (2012) Consistency in multi-vehicle inventory-routing. Transp Res C-EMER 24:270–287CrossRefGoogle Scholar
- Coelho LC, Cordeau JF, Laporte G (2014) Thirty years of inventory-routing. Transp Sci 48(1):1–19CrossRefGoogle Scholar
- Cui J, Engell S (2010) Medium-term planning of a multiproduct batch plant under evolving multi-period multi-uncertainty by means of a moving horizon strategy. Comput Chem Eng 34(5):598–619CrossRefGoogle Scholar
- Dong Y, Pinto JM, Sundaramoorthy A, Maravelias CT (2014) MIP model for inventory routing in industrial gases supply chain. Ind Eng Chem Res 53(44):17214–17225CrossRefGoogle Scholar
- Dong Y, Maravelias CT, Pinto JM, Sundaramoorthy A (2017) Solution methods for vehicle-based inventory routing problems. Comput Chem Eng 101:259–278CrossRefGoogle Scholar
- Engineer FG, Furman KC, Nemhauser GL, Savelsbergh MWP, Song J-H (2012) A branch-price-and-cut algorithm for single-product maritime inventory routing. Oper Res 60(1):106–122MathSciNetCrossRefMATHGoogle Scholar
- Fischer A, Nokhart H, Olsen H, Fagerholt K, Rakke JG, Stålhane M (2016) Robust planning and disruption management in roll-on roll-off liner shipping. Transp Res E-LOG 91:51–67CrossRefGoogle Scholar
- Gaur V, Fisher ML (2004) A periodic inventory routing problem at a supermarket chain. Oper Res 52(6):813–822CrossRefMATHGoogle Scholar
- Goel V, Furman KC, Song JH, El-Bakry AS (2012) Large neighborhood search for LNG inventory routing. J Heuristics 18(6):821–848CrossRefGoogle Scholar
- Golden BL, Raghavan S, Wasil EA (2008) The vehicle routing problem: latest advances and new challenges, vol 43. Springer, BerlinCrossRefMATHGoogle Scholar
- Gounaris CE, Wiesemann W, Floudas CA (2013) The robust capacitated vehicle routing problem under demand uncertainty. Oper Res 61(3):677–693MathSciNetCrossRefMATHGoogle Scholar
- Grønhaug R, Christiansen M, Desaulniers G, Descrosiers J (2010) A branch-and-price method for a liquefied natural gas inventory routing problem. Transp Sci 44(3):400–415CrossRefGoogle Scholar
- Gupta D, Maravelias CT (2016) On deterministic online scheduling: major considerations, paradoxes and remedies. Comput Chem Eng 94:312–330CrossRefGoogle Scholar
- Gupta D, Maravelias CT (2017) A general state-space formulation for online scheduling. Processes 5(4):69CrossRefGoogle Scholar
- Gupta D, Maravelias CT, Wassick JM (2016) From rescheduling to online scheduling. Chem Eng Res Des 116:83–97CrossRefGoogle Scholar
- Harjunkoski I, Maravelias CT, Bongers P, Castro PM, Engell S, Grossmann IE, Hooker J, Méndez C, Sand G, Wassick J (2014) Scope for industrial applications of production scheduling models and solution methods. Comput Chem Eng 62:161–193CrossRefGoogle Scholar
- Janak SL, Lin X, Floudas CA (2007) A new robust optimization approach for scheduling under uncertainty: II. Uncertainty with known probability distribution. Comput Chem Eng 31(3):171–195CrossRefGoogle Scholar
- Jiang Y, Grossmann IE (2015) Alternative mixed-integer linear programming models of a maritime inventory routing problem. Comput Chem Eng 77:147–161CrossRefGoogle Scholar
- Kleywegt AJ, Nori VS, Savelsbergh MWP (2002) The stochastic inventory routing problem with direct deliveries. Transp Sci 36(1):94–118CrossRefMATHGoogle Scholar
- Laporte G (2009) Fifty years of vehicle routing. Transp Sci 43(4):408–416CrossRefGoogle Scholar
- Laporte G, Gendreau M, Potvin JY, Semet F (2000) Classical and modern heuristics for the vehicle routing problem. Int Trans Oper Res 7(4–5):285–300MathSciNetCrossRefGoogle Scholar
- Li Z, Ierapetritou M (2008) Process scheduling under uncertainty: review and challenges. Comput Chem Eng 32:715–727CrossRefGoogle Scholar
- Mastragostino R, Patel S, Swartz CLE (2014) Robust decision making for hybrid process supply chain systems via model predictive control. Comput Chem Eng 62(5):37–55CrossRefGoogle Scholar
- Méndez CA, Cerdá J, Grossmann IE, Harjunkoski I, Fahl M (2006) State-of-the-art review of optimization methods for short-term scheduling of batch processes. Comput Chem Eng 30(6):913–946CrossRefGoogle Scholar
- Mestan E, Türkay M, Arkun Y (2006) Optimization of operations in supply chain systems using hybrid systems approach and model predictive control. Ind Eng Chem Res 45(19):6493–6503CrossRefGoogle Scholar
- Moin NH, Salhi S (2007) Inventory routing problems: a logistical overview. J Oper Res Soc 58(9):1185–1194CrossRefMATHGoogle Scholar
- Nandola NN, Rivera DE (2013) An improved formulation of hybrid model predictive control with application to production-inventory systems. IEEE Trans Control Syst Technol 21(1):121–135CrossRefGoogle Scholar
- Novas JM, Henning GP (2010) Reactive scheduling framework based on domain knowledge and constraint programming. Comput Chem Eng 34(12):2129–2148CrossRefGoogle Scholar
- Ortega M, Lin L (2004) Control theory applications to the production-inventory problem: a review. Int J Prod Res 42:2303–2322CrossRefMATHGoogle Scholar
- Ouelhadj D, Petrovic S (2009) A survey of dynamic scheduling in manufacturing systems. J Sched 12:417–431MathSciNetCrossRefMATHGoogle Scholar
- Papageorgiou DJ, Nemhauser GL, Sokol J, Cheon M-S, Keha AB (2014) MIRPLib—a library of maritime inventory routing problem instances: survey, core model, and benchmark results. Eur J Oper Res 235(2):350–366MathSciNetCrossRefMATHGoogle Scholar
- Papageorgiou DJ, Cheon MS, Harwood S, Trespalacios F, Nemhauser GL (2018) Recent progress using matheuristics for strategic maritime inventory routing. In: Konstantopoulos C, Pantziou G (eds) Modeling, computing and data handling methodologies for maritime transportation, Springer, Cham, pp 59–94CrossRefGoogle Scholar
- Perea-López E, Ydstie BE, Grossmann IE (2003) A model predictive control strategy for supply chain optimization. Comput Chem Eng 27(8):1201–1218CrossRefGoogle Scholar
- Pinedo M (2012) Scheduling: theory, algorithms, and systems. Springer, BerlinCrossRefMATHGoogle Scholar
- Pochet Y, Wolsey LA (2006) Production planning by mixed integer programming. Springer, BerlinMATHGoogle Scholar
- Rakke JG, Stålhane M, Moe CR, Christiansen M, Andersson H, Fagerholt K, Norstad I (2011) A rolling horizon heuristic for creating a liquefied natural gas annual delivery program. Transp Res C-EMER 19(5):896–911CrossRefGoogle Scholar
- Ronen D (2002) Marine inventory routing: shipments planning. J Oper Res Soc 53(1):108–114CrossRefMATHGoogle Scholar
- Sahin F, Robinson EP, Gao L-L (2008) Master production scheduling policy and rolling schedules in a two-stage make-to-order supply chain. Int J Prod Econ 115(2):528–541CrossRefGoogle Scholar
- Sarimveis H, Patrinos P, Tarantilis CD, Kiranoudis CT (2008) Dynamic modeling and control of supply chains: a review. Comput Oper Res 35:3530–3561CrossRefMATHGoogle Scholar
- Singh T, Arbogast JE, Neagu N (2015) An incremental approach using local-search heuristic for inventory routing problem in industrial gases. Comput Chem Eng 80:199–210CrossRefGoogle Scholar
- Siswanto N, Essam D, Sarker R (2011) Solving the ship inventory routing and scheduling problem with undedicated compartments. Comput Ind Eng 61(2):289–299CrossRefGoogle Scholar
- Song J-H, Furman KC (2013) A maritime inventory routing problem: practical approach. Comput Oper Res 40(3):657–665CrossRefMATHGoogle Scholar
- Subramanian K, Maravelias CT, Rawlings JB (2012) A state-space model for chemical production scheduling. Comput Chem Eng 47:97–110CrossRefGoogle Scholar
- Subramanian K, Rawlings JB, Maravelias CT, Flores-Cerrillo J, Megan L (2013) Integration of control theory and scheduling methods for supply chain management. Comput Chem Eng 51:4–20CrossRefGoogle Scholar
- Subramanian K, Rawlings JB, Maravelias CT (2014) Economic model predictive control for inventory management in supply chains. Comput Chem Eng 64:71–80CrossRefGoogle Scholar
- Toth P, Vigo D (2001) The vehicle routing problem. Society for Industrial and Applied Mathematics, PhiladelphiaMATHGoogle Scholar
- Verderame PM, Elia JA, Li J, Floudas CA (2010) Planning and scheduling under uncertainty: a review across multiple sectors. Ind Eng Chem Res 49(9):3993–4017CrossRefGoogle Scholar
- Vieira G, Herrmann JW, Lin E (2003) Rescheduling manufacturing systems: a framework of strategies, policies, and methods. J Sched 6:39–62MathSciNetCrossRefMATHGoogle Scholar
- Wonnacott TH, Wonnacott RJ (1990) Introductory statistics for business and economics. Wiley, TorontoMATHGoogle Scholar
- Zhang Q, Sundaramoorthy A, Grossmann IE, Pinto JM (2017) Multiscale production routing in multicommodity supply chains with complex production facilities. Comput Oper Res 79:207–222MathSciNetCrossRefGoogle Scholar
- Zhang C, Nemhauser G, Sokol J, Cheon MS, Keha A (2018) Flexible solutions to maritime inventory routing problems with delivery time windows. Comput Oper Res 89:153–162MathSciNetCrossRefGoogle Scholar