A robust optimization model for the maritime inventory routing problem

Abstract

Uncertainty is a highly important aspect of maritime transportation. Unforeseen occurrences related to environmental conditions, poor weather, vessel reliability, or port congestion are frequent and have a non-negligible impact on the total time required for vessels to perform (un)loading operations at ports. We study a special case of maritime transportation named the maritime inventory routing (MIR) problem, in which one must determine the routings of vessels while keeping the inventory levels at ports within the operational limits. In this paper, we propose a robust optimization approach that considers the uncertainty in the total time spent by vessels at the ports. This approach allows the trade-off between the risk of infeasibility (i.e., violating inventory limits at ports) and the increase in operational costs due to the protection against uncertainty events to be assessed. To test the proposed methodology, we used a real-world instance based on the MIR problem faced by a Brazilian petroleum company. In this problem, violating the inventory limits at ports causes considerable financial losses due to consequent interruptions in crude oil production. Our approach supports the decision maker to devise more robust plans in which the risk of violating inventory limits is acceptable. In other words, despite the increase in the operational costs associated with more robust solutions, the approach enables the decision maker to avoid much larger potential costs. For the problem considered, we observed that the probability of infeasibility of the proposed solution may be reduced from 87% to 2%, depending on the level of robustness adopted by the decision maker. However, this increased protection causes an increase of up to 13% in the overall costs.

Appendices

Appendix 1

Here we present a list of symbols used in the formulations in this paper. Sets and parameters are represented by capital letters and variables are in lower case. Some symbols may be represented by more than one letter for the sake of readability. The first part of symbols is used in the FCNF formulation.

$N$ :

Set of all ports indexed by $i$ and $j$

$T$ :

Set of time periods indexed by $t$

$V$ :

Set of vessels indexed by $v$

$NP$ :

Set of loading ports indexed by $i$ and $j$

$ND$ :

Set of discharge ports indexed by $i$ and $j$

$C_{ijv}^{T}$ :

Sailing cost of vessel $v$ between ports $i$ and $j$

$C_{v}^{W}$ :

Waiting cost of vessel $v$ for each time period

$C_{iv}^{P}$ :

Port cost of port $i$ for vessel $v$

$O_{v}$ :

The position of vessel $i$ in the beginning of planning horizon

$DE_{v}$ :

The artificial end node for each vessel $v$

$T_{ijv}$ :

Sailing time of vessel $v$ between ports $i$ and $j$

$B_{it}$ :

Number of berths available in port $i$ during time period $t$

$Q_{v}$ :

The maximum amount of product to be loaded at one time period of ship $v$

$L0_{v}$ :

Inventory on board of vessel $v$ in the beginning of the planning horizon

$K_{v}$ :

Vessel capacity

$D_{it}$ :

Demand rate in the discharge port $i$ for each time period

$P_{it}$ :

Production rate in the load port $i$ for each time period

$SMX_{it}$ :

Upper bound of inventory level in port $i$ for each time period

$SMN_{it}$ :

Lower bound of inventory level in port $i$ for each time period

$S0_{i}$ :

Inventory level at port $i$ in the beginning of the planning horizon

$o_{ivt}$ :

Where $o_{ivt}$ ∈ {0,1} equal to 1 if the vessel is operating (loading/discharging) in port during time period $t$ and 0 otherwise,

$x_{ijvt}$ :

Where $x_{ijvt}$∈ {0,1} equal to 1 if vessel $v$ left port $i$ to port $j$ during time period $t$ and 0 otherwise,

$w_{ivt}$ :

Where $w_{ivt}$ ∈ {0,1} equal to 1 if vessel $v$ is waiting outside berth in port $i$ during time period $t$ and 0 otherwise,

$l_{vt}$ :

Where $l_{vt}$ ∈ R inventory level on board vessel $v$ during time period $t$,

$q_{ivt}$ :

Where $q_{ivt}$ ∈ R quantity (un)load from/to vessel $v$ in port $i$ during time period $t$,

$s_{it}$ :

Where $s_{it}$∈ R inventory level at port $i$ during time period $t$

$o_{ivt}^{A}$ :

Where $o_{ivt}^{A}$ ∈ {0,1} indicates whether ship $v$ starts to operate at port $i$ in period $t$

$o_{ivt}^{B}$ :

Where $o_{ivt}^{B}$ ∈ {0,1} indicates whether ship $v$ continue to operate at port $i$ in period $t$

$f_{ijvt}^{X}$ :

That indicates the load on board ship $v$ when traveling from port $i$ to port $j$, leaving port $i$ in period $t$,

$f_{ivt}^{OA}$ :

That indicates the load on board ship $v$ when starting to operate at port $i$ in period $t$ and has not operated in period $t - 1$,

$f_{ivt}^{OB}$ :

That indicates the load on board ship $v$ before continuing to operate at port $i$ in period $t$ after having operated in time period $t - 1$,

$f_{ivt}^{W}$ :

That indicates the load on board ship $v$ while waiting during time period $t$ at port $i$

The following symbols are used to reformulate FCNF model to become RO-FCNF model.

$K_{iv}$ :

Set of coefficients $OH_{iv}$ that are subject to parameter uncertainty

UNB(v) :

Subset of T that comprehend the time periods in the beginning of the planning horizon when each vessel is unavailable

UNE(v) :

Subset of T that comprehends the time periods at the end of planning horizon when each vessel is unavailable

TT :

The last time period of the planning horizon T,

$HR_{v}$ :

The hire fee paid in a daily basis to vessels owner,

$OH_{iv}$ :

Nominal time spent by vessel v at port i,

$OHD_{iv}$ :

Maximum deviation observed for the time spent by vessel v at port i,

$\widetilde{OH}_{iv}$ :

Random variable representing the time spent by vessel v at port i,

$\varGamma_{iv}$ :

Represents the maximum number of coefficients parameters of constraints (i,v) that can deviate from their nominal value. It adjusts the robustness of the decision. The parameter $\varGamma_{iv} \in \left[ {0,\left| {K_{iv} } \right|} \right]$. Hence, the parameter $\varGamma_{iv}$ protects against deviations in up to $\varGamma_{iv}$ of these coefficients. In other words, we stipulate that nature will be restricted in its behavior, in that only a subset of the coefficients will change in order to adversely affect the solution

$nh_{v}^{NT}$ :

Binary variable that indicates whether a vessel stayed idle during the entire planning horizon or not

$k_{ijvt}$ :

Auxiliary variable representing the scaled deviation of time spent at port $k_{ijvt} = \left( {\widetilde{OH}_{iv} - OH_{iv} } \right)/OHD_{iv}$,

$\pi_{iv}$, $\rho_{ijvt}$:

Dual variables

Appendix 2

Next, we present FCNF model, first proposed by Agra et al. (2013).

Objective function:

$$min \mathop \sum \limits_{v \in V} \mathop \sum \limits_{{i \in N\left\{ {O_{v} } \right\}}} \mathop \sum \limits_{{j \in N \cup \left\{ {DE_{v} } \right\}}} \mathop \sum \limits_{t \in T} C_{ijv}^{T} .x_{ijvt} + \mathop \sum \limits_{v \in V} \mathop \sum \limits_{i \in N} \mathop \sum \limits_{t \in T} C_{iv}^{P} .x_{ijvt} + \mathop \sum \limits_{v \in V} \mathop \sum \limits_{i \in N} \mathop \sum \limits_{t \in T} C_{v}^{W} .w_{ivt}$$

(1)

The minimization function (1) contains transportation costs, operation costs and waiting costs.Constraints

$$\mathop \sum \limits_{{j \in N \cup \left\{ {DE_{v} } \right\}}} \mathop \sum \limits_{t \in T} x_{{O_{v} jvt}} = 1, \quad \forall v \in V,$$

(2)

$$\mathop \sum \limits_{{i \in N \cup \left\{ {O_{v} } \right\}}} \mathop \sum \limits_{t \in T} x_{{iDE_{v} vt}} = 1, \quad \forall v \in V,$$

(3)

$$o_{iv,t - 1} \le \mathop \sum \limits_{{j \in N \cup \left\{ {DE_{v} } \right\}}} x_{ijvt} + o_{ivt} , \quad \forall v \in V, i \in N, t \in T,$$

(4)

$$o_{iv,t - 1} \ge \mathop \sum \limits_{{j \in N \cup \left\{ {DE_{v} } \right\}}} x_{ijvt} , \quad \forall v \in V, i \in N, t \in T,$$

(5)

Constraints (2) and (3) guarantee that every ship leaves from its artificial origin port and finishes the voyage at its artificial destination port. Constraints (4) determine that after an operation period, a vessel can continue to operate or sail the port in the next period and constraints (5) ensure that before leaving a port, the vessel must have operated there in the time period immediately before.

$$\mathop \sum \limits_{{j \in N \cup \left\{ {O_{v} } \right\}}} x_{{jiv,t - T_{jiv} }} + w_{iv,t - 1} = w_{ivt} + o_{ivt }^{A} , \quad \forall v \in V, i \in N, t \in T,$$

(6)

$$o_{ivt - 1}^{A} + o_{ivt - 1}^{B} = o_{ivt}^{B} + \mathop \sum \limits_{{j \in N \cup \left\{ {DE_{v} } \right\}}} x_{ijvt} , \quad \forall v \in V, i \in N, t \in T,$$

(7)

$$o_{ivt}^{A} + o_{ivt}^{B} = o_{ivt} , \quad \forall v \in V, i \in N, t \in T,$$

(8)

$$\mathop \sum \limits_{v \in V} o_{ivt} \le B_{it} , \quad \forall i \in N, t \in T,$$

(9)

Constraints (6) and (7) represent vessel’s movement along the time–space network structure. At every time period, the vessel must be leaving a port, waiting to operate or operating at a port. These constraints also connect the time when a vessel leaves one port $j$ to the time when the same vessel arrives at a port $i$. Constraints (8) provide the coordination between the path of the ships and the loading or discharging of crude oil in a port. Constraints (9) give berth restriction at each node and consequently waiting time.

$$0 \le q_{ivt} \le Q_{v} o_{ivt} , \quad \forall v \in V, i \in N, t \in T,$$

(10)

$$\mathop \sum \limits_{{j \in N \cup \left\{ {O_{v} } \right\}}} f_{{jiv,t - T_{jiv} }}^{X} + f_{iv,t - 1}^{W} = f_{ivt}^{W} + f_{ivt}^{OA} ,\quad \forall v \in V, i \in N, t \in T,$$

(11)

$$f_{iv,t - 1}^{OA} + f_{iv,t - 1}^{OB} + q_{iv,t - 1} = f_{ivt}^{OB} + \mathop \sum \limits_{{j \in N \cup \left\{ {DE_{v} } \right\}}} f_{ijvt}^{X} ,\quad \forall v \in V, i \in NP \cup \left\{ {O_{v} } \right\}, t \in T,$$

(12)

$$f_{iv,t - 1}^{OA} + f_{iv,t - 1}^{OB} - q_{iv,t - 1} = f_{ivt}^{OB} + \mathop \sum \limits_{{j \in N \cup \left\{ {DE_{v} } \right\}}} f_{ijvt}^{X} ,\quad \forall v \in V, i \in ND \cup \left\{ {O_{v} } \right\}, t \in T,$$

(13)

$$f_{{O_{v} jvt}}^{X} = L0_{v}^{{}} .x_{{O_{v} jvt}} ,\quad \forall v \in V, j \in N \cup \left\{ {DE_{v} } \right\}, t \in T$$

(14)

The quantity to be (un)loaded ($q_{ivt}$) must be less than the maximum amount of product to be loaded or discharged from ship v (constraints 10). The load onboard the vessel on each arc of the network flow is controlled in constraints (11–13). Initial inventory level onboard the vessel is given in constraints (14).

$$0 \le f_{ijvt}^{X} \le K_{v} .x_{ijvt} ,\quad \forall v \in V, i \in N \cup \left\{ {O_{v} } \right\}, j \in N \cup \left\{ {DE_{v} } \right\}, t \in T$$

(15)

$$0 \le f_{ivt}^{OA} \le K_{v} .o_{ivt}^{A} , \quad \forall v \in V, i \in N, t \in T,$$

(16)

$$0 \le f_{ivt}^{OB} \le K_{v.} o_{ivt}^{B} ,\quad \forall v \in V, i \in N, t \in T,$$

(17)

$$0 \le f_{ivt}^{W} \le K_{v} .w_{ivt} ,\quad \forall v \in V, i \in N, t \in T.$$

(18)

The variables upper bounds (vessel capacity) and non-negativity constraints are expressed in (15–18):

$$s_{i,t - 1} + \mathop \sum \limits_{v \in V} q_{ivt} = D_{it} + s_{it} , \quad \forall i \in ND, t \in T,$$

(19)

$$s_{i,t - 1} + P_{it} = \mathop \sum \limits_{v \in V} q_{ivt} + s_{it} , \quad \forall i \in NP, t \in T,$$

(20)

$$SMN_{it} \le s_{it} \le SMX_{it} , \quad \forall i \in N, t \in T,$$

(21)

$$s_{i0} = S0_{i}^{{}} , \quad \forall i \in N,$$

(22)

The inventory level at each port is controlled during every time period of the planning horizon at load and discharge ports (constraints 19 and 20). Constraints (21) give the operational range for inventory levels and constraints (22) define initial inventory levels at each port.

2.1 Reformulations proposed

The first reformulation that we propose represents the fixed costs of the fleet. To achieve that, we add hiring costs to the objective function and formulate constraints that ensure whether the vessel is used during that planning horizon.

Objective function

$$min \mathop \sum \limits_{v \in V} \mathop \sum \limits_{{i \in N \cup \left\{ {O_{v} } \right\}}} \mathop \sum \limits_{{j \in N \cup \left\{ {DE_{v} } \right\}}} \mathop \sum \limits_{t \in T} C_{ijv}^{T} .x_{ijvt} + \mathop \sum \limits_{v \in V} \mathop \sum \limits_{i \in N} \mathop \sum \limits_{t \in T} C_{iv}^{P} x_{ijvt} + \mathop \sum \limits_{v \in V} \mathop \sum \limits_{i \in N} \mathop \sum \limits_{t \in T} C_{v}^{W} w_{ivt} + \mathop \sum \limits_{{\varvec{v} \in \varvec{V}}} (1 - \varvec{nh}_{\varvec{v}}^{{\varvec{NT}}} )\varvec{HR}_{\varvec{v}} \varvec{TT}$$

(23)

Constraints

$$nh_{v}^{NT} = \mathop \sum \limits_{{i = O_{v} }} \mathop \sum \limits_{{j = DE_{v} }} \mathop \sum \limits_{t \in T} x_{ijvt} \quad \forall v \in V$$

(24)

In (23), we add (in bold) the fixed costs for every vessel that is utilized during the planning horizon to the original objective function. Note that regardless of whether the vessel is utilized for one day or the entire planning horizon, the hire fee is charged for the complete planning horizon. Constraints (24) ensure that if a vessel goes directly from the artificial origin node to the artificial destination node, then it stays idle for the entire planning horizon.

$$\mathop \sum \limits_{t \in T} o_{ivt} \ge \mathop \sum \limits_{t \in T,j \in N} OH_{iv} .x_{ijvt} , \quad \forall i \in N,v \in V,$$

(25)

Constraints (25) determine that the number of time-periods a vessel v spends at port i must be at least the product of the nominal time spent at port $OH_{i,v}$ and the number of visits of vessel v at port i. For example, if the nominal time spent at port i is equal to 1 day and vessel v visits the port 3 times, it means that the sum of operation time $\mathop \sum \limits_{t \in T} o_{ivt}$ (or time spent at port) of the vessel v at port i during the entire planning horizon must be equal to or higher than 3 (the product of the nominal time spent at port i and the number of visits of vessel v at port i).

Appendix 3

To consider uncertainty in the time spent at ports (parameter $OH_{iv}$), we reformulate constraints (25) into a stochastic version (25s), where $OH_{iv}$ may deviate up to its maximum deviation $OHD_{iv}$ according to its scalar deviation $k_{ijvt}$:

$$\mathop \sum \limits_{t \in T} o_{ivt} \ge \mathop \sum \limits_{t \in T,j \in N} OH_{i,v} .x_{ijvt} + \mathop \sum \limits_{t \in T,j \in N} OHD_{iv} .x_{ijvt} .k_{ijvt} ,\quad \forall i \in N,v \in V$$

(25s)

In order to build the robust counterpart of the model, it is necessary to reformulate constraints (25s). First, we adopt the robust paradigm, which considers that uncertainty will behave as worse as possible. In other words, (25 s) indicates that the solution would be feasible for every realization of the time spent at port.

$$\varPhi \left( {x,\varGamma } \right) = Max_{k} \left\{ \mathop \sum \limits_{t \in T,j \in N} OHD_{iv} .x_{ijvt} .k_{ijvt} |\mathop \sum \limits_{t \in T,j \in N} k_{ijvt} \le \varGamma_{iv} ,k_{ijvt} \ge 0\right\}$$

(26)

Note that (26) protects against the uncertainty for every visit of a vessel v at a port i and it is known as protection function. We disregard $k_{ijvt} \in \left[ { - 1,0} \right]$, because we intend to protect only against the worst cases. Moreover, the parameter $\varGamma_{iv}$, introduced to adjust the model robustness against the conservatism of the solution, can be understood as the maximum number of the uncertain parameters that can deviate from their nominal values. $\varGamma_{iv}$ may take values in the interval [0,|K|], where |K| represents the maximum number of visits a port i may receive during the entire planning horizon.

The robust counterpart of (25) is

$$\mathop \sum \limits_{t \in T} o_{ivt} \ge \mathop \sum \limits_{t \in T,j \in N} OH_{i,v} .x_{ijvt} + \varPhi \left( {x,\varGamma } \right), \quad \forall i \in N,v \in V$$

(27)

Applying the robust optimization technique developed by Bertsimas and Sim (2004), we formulate the auxiliary problem (28–30). Its objective is to maximize the sum of all deviations over the set of all admissible realizations of the uncertain parameters.

$$Max_{k} \mathop \sum \limits_{t \in T,j \in N} OHD_{iv} .x_{ijvt}^{*} .k_{ijvt}$$

(28)

Subject to:

$$\mathop \sum \limits_{j,t} k_{ijvt} \le \Gamma_{iv} ,\quad \forall i,v$$

(29)

$$0 \le k_{ijvt} \le 1,\quad \forall i,j,v,t$$

(30)

If $\Gamma_{iv} = 0$, $k_{ijvt}$, for all i,j,v,t, is forced to be 0, so that the random variable $\widetilde{OH}_{iv}$ is equal to their mean value $OH_{i,v}$ and there is no protection against uncertainty. On the other hand, when $\Gamma_{iv} =$|K|, $k_{ijvt}$, for all i,j,v,t, is forced to be 1 (in this particular problem) and constraints (25 s) is completely protected against uncertainty, yielding a very conservative solution. For values between 0 and |K|, the decision-maker can tradeoff between the protection level and the degree of conservatism of the solution.

Following the same rationale of Bertsimas and Sim (2004), the dual of the model (28–30) is stated as follows:

$$Min_{\pi ,\rho } \pi_{iv} .\Gamma_{iv} + \mathop \sum \limits_{t \in T,j \in N} \rho_{ijvt} ,\quad \forall i,v$$

(31)

Subject to:

$$\pi_{iv} + \rho_{ijvt} \ge OHD_{iv} .x_{ijvt} ,\quad \forall i,j,v,t$$

(32)

$$\rho_{ijvt} \ge 0,\quad\forall i,j,v,t$$

(33)

$$\pi_{iv} \ge 0, \quad \forall i,v$$

(34)

This dual problem has two dual variables ($\pi_{iv}$, $\rho_{ijvt}$) that are associated to constraints (28) and (30), respectively. By strong duality, as model (28–30) is feasible and bounded for all $\Gamma_{iv}$ ∈ [0,|K_iv|], then the dual problem (31–34) is also feasible and their objective function values coincide.

Substituting (31–34) in constraints (27), the following robust linear set of constraints are obtained.

$$\mathop \sum \limits_{t \in T,j \in N} OH_{iv} .x_{ijvt} + \pi_{iv} .\Gamma_{iv} + \mathop \sum \limits_{t \in T,j \in N} \rho_{ijvt} - \mathop \sum \limits_{t \in T} o_{ivt} \le 0,\quad \forall i \in N,v \in V,$$

(35)

$$\pi_{iv} + \rho_{ijvt} \ge OHD_{iv} .x_{ijvt} ,\quad\forall i,j,v,t$$

(36)

$$\rho_{ijvt} \ge 0,\quad\forall i,j,v,t$$

(37)

$$\pi_{iv} \ge 0,\quad\forall i,v$$

(38)

Adding (35–38) to the original FCNF formulation, we have now the RO-FCNF formulation. This model minimizes transportation costs and ensures that up to $\Gamma$ coefficients deviate their value from the mean time spent at a port within the permitted interval ($\left[ {OH_{iv} - OHD_{iv} ,OH_{iv} + OHD_{iv} } \right]$), then the solution of the robust optimization model will remain feasible. In other words, the solution of this model is a robust solution.

Appendix 4

Next, we present additional constraints in the deterministic FCNF model used to set vessel availability for each level of Γ.

$$\mathop \sum \limits_{{t \in UNB_{v} }} x_{ijvt} = 0 \quad \forall i,j \in N, \quad \forall v \in V$$

(39)

$$\mathop \sum \limits_{{t \in UNE_{v} }} x_{ijvt} = 0 \quad \forall i,j \in N, \quad \forall v \in V$$

(40)

Constraints (39) and (40) define the period of vessel unavailability in the beginning and at the end of the planning horizon, respectively. It is important to highlight that constraint (39, 40) are only used in the simulation phase of the robust optimization approach, because the unavailability periods (in the beginning and at the end of planning horizon) are only known after the robust model is solved for each level of Γ during the second phase of the methodology.