Recent Progress Using Matheuristics for Strategic Maritime Inventory Routing

Abstract

This chapter presents an extensive computational study of simple, but prominent matheuristics (i.e., heuristics that rely on mathematical programming models) to find high quality ship schedules and inventory policies for a class of maritime inventory routing problems. Our computational experiments are performed on a test bed of the publicly available MIRPLib instances. This class of inventory routing problems has few constraints relative to some operational problems, but is complicated by long planning horizons. We compare several variants of rolling horizon heuristics, K-opt heuristics, local branching, solution polishing, and hybrids thereof. Many of these matheuristics substantially outperform the commercial mixed-integer programming solvers CPLEX 12.6.2 and Gurobi 6.5 in their ability to quickly find high quality solutions. New best known incumbents are found for 26 out of 70 yet-to-be-proved-optimal instances and new best known bounds on 56 instances.

Appendices

3.6 Nomenclature

Sets are denoted using capital letters in a calligraphic font, such as $\mathcal {T}$ and $\mathcal {V}$. Parameters are typically denoted with capital letters in italic font or with Greek characters. Decision variables always appear as lower case letters.

Indices and Sets

$$ \begin{array}{ll} t \in \mathcal {T} &{} \text {set of time periods with} T = |\mathcal {T}|\\ v \in \mathcal {V} &{} \text {set of vessels} \\ vc \in \mathcal {VC} &{} \text {set of vessel classes} \\ j \in \mathcal {J}^P &{} \text {set of production, a.k.a. loading, ports} \\ j \in \mathcal {J}^C &{} \text {set of consumption, a.k.a. discharging, ports} \\ j \in \mathcal {J} &{} \text {set of all ports:}\, \mathcal {J} = \mathcal {J}^P \cup \mathcal {J}^D \\ n \in \mathcal {N} &{} \text {set of regular nodes or port-time pairs:}\, \mathcal {N} = \{ n = (j,t): j \in \mathcal {J}, t \in \mathcal {T} \} \\ n \in \mathcal {N}_{0,T+1} &{} \text {set of all nodes (including a source node}\, n_{_{0}}\,\text {and a sink node}\,{n_{_{T+1}}}) \\ a \in \mathcal {A} &{} \text {set of all arcs} \\ a \in \mathcal {A}^v &{} \text {set of arcs associated with vessel}\, v\, \in \mathcal {V} \\ a \in \mathcal {A}^{vc} &{} \text {set of arcs associated with vessel class}\, vc\, \in \mathcal {VC} \\ a \in \mathcal {FS}_n^{vc} &{} \text {forward star associated with node}\, n = (j,t) \in \mathcal {N}_{s,t}\,\text {and vessel class}\, vc \in \mathcal {VC} \\ a \in \mathcal {RS}_n^{vc} &{} \text {reverse star associated with node}\, n = (j,t) \in \mathcal {N}_{s,t}\,\text {and vessel class}\, vc \in \mathcal {VC} \\ \end{array} $$

Data

$$ \begin{array}{ll} B_j &{} \text {number of berths (berth limit) at port}\, j\, \in \mathcal {J} \\ C_a^{vc} &{} \text {cost for vessel class vc to traverse arc}\, a = ((j_1,t_1),(j_2,t_2)) \in \mathcal {A}^{vc} \\ D_{j,t} &{} \text {number of units produced/consumed at port}\,j \,\in \mathcal {J}\, \text {in period}\, t \in \mathcal {T} \\ \Delta _j &{} \text {an indicator parameter taking value} +1 \text {if}\, j\, \in \mathcal {J}^P \text {and}\, -1 \text {if}\, j \in \mathcal {J}^C \\ &{} ~~~\text {at port}\, j \,\text {from a single vessel in a period} \\ P_{j} &{} \text {penalty for unmet demand/excess inventory at port}\,\,j\, \in \mathcal {J}\,\,\text {in time period}\, t\, \in \mathcal {T} \\ Q^v (Q^{vc}) &{} \text {capacity of vessel}\, v\, \in \mathcal {V}\,\text {(capacity of a vessel in vessel class}\, vc) \\ S_{j,t}^{\min } (S_{j,t}^{\max }) &{} \text {lower bound (capacity) at port}\, j\, \in \mathcal {J}\,\,\text {in time period}\, t \in \mathcal {T} \\ s_{j,0} &{} \text {initial inventory at port}\, j\, \in \mathcal {J} \\ \end{array} $$

Decision Variables

$$ \begin{array}{ll} \alpha _{j,t} &{} \text {(continuous) number of units of unmet demand/excess inventory at port}\, j\, \in \mathcal {J}\,\text {in time period}\, t\, \in \mathcal {T} \\ s_{j,t} &{} \text {(continuous) number of units of inventory at port}\, j\, \in \mathcal {J}\,\text {available\,at\,the}\,end \,\text {of period}\, t\, \\ x_a^{vc} &{} \text {(integer) number of vessels in vessel class}\, vc\, \in \mathcal {VC}\,\,\text {using arc}\,\, a \in \mathcal {A}^{vc}. \\ \end{array} $$

Appendix

3.7 Appendix

Lemma 3.1

Consider the 2-variable mixed-integer linear sets

$$ \mathcal {S}_{\le } = \left\{ (x,y) \in \mathbb {Z}\times \mathbb {R}_+ : x - y \le b \right\} ~. $$

and

$$ \mathcal {S}_{\ge } = \left\{ (x,y) \in \mathbb {Z}\times \mathbb {R}_+ : x + y \ge b \right\} ~. $$

Then, the inequality

$$\begin{aligned} x - \frac{1}{1-f_0} y \le \lfloor b \rfloor \end{aligned}$$

(3.8)

is valid for $\mathrm{conv}(\mathcal {S}_{\le })$ where $f_0 := b - \lfloor b \rfloor$. The inequality

$$\begin{aligned} x + \frac{1}{1-f_0} y \ge \lceil b \rceil \end{aligned}$$

(3.9)

is valid for $\mathrm{conv}(\mathcal {S}_{\ge })$ where $f_0 := \lceil b \rceil - b$.

Proof

See [16].    $\square$

Consider a mixed-integer linear set defined by a single constraint:

$$ \mathcal {S} = \left\{ (\mathbf {x},\mathbf {y}) \in \mathbb {Z}_+^n \times \mathbb {R}_+^p : \mathbf {a}^{\top }\mathbf {x} + \mathbf {g}^{\top }\mathbf {y} \le \mathbf {b} \right\} ~. $$

Proposition 3.1

The Mixed-Integer Rounding (MIR) inequality

$$\begin{aligned} \sum _{j=1}^n \left( \lfloor a_j \rfloor + \frac{(f_j-f_0)^+}{1-f_0} \right) x_j + \frac{1}{1-f_0} \sum _{j:g_j < 0} g_j y_j \le \lfloor b \rfloor \end{aligned}$$

(3.10)

is valid for conv($\mathcal {S}$) where $f_0 = b - \lfloor b \rfloor$ and $f_j = a_j - \lfloor a_j \rfloor$.¹ When the single constraint in $\mathcal {S}$ is written with a $\ge$ sign, the MIR inequality becomes

$$\begin{aligned} \sum _{j=1}^n \left( \lceil a_j \rceil - \frac{(f_j-f_0)^+}{1-f_0} \right) x_j + \frac{1}{1-f_0} \sum _{j:g_j > 0} g_j y_j \ge \lceil b \rceil \end{aligned}$$

(3.11)

where $f_0 = \lceil b \rceil - b$ and $f_j = \lceil a_j \rceil - a_j$.

Proof

Since the proof of the first case with $\mathcal {S}$ expressed with a $\le$ sign is given in [16], we prove the second case. Note that $a_j = \lceil a_j \rceil - f_j = \lfloor a_j \rfloor + (1-f_j)$. Relax the constraint $\sum _{j=1}^n a_j x_j + \sum _{j=1}^p g_j y_j \ge b$ to

$$\begin{aligned} \sum _{j:f_j \le f_0} \lceil a_j \rceil x_j + \sum _{j:f_j> f_0} a_j x_j + \sum _{j:g_j > 0} g_j y_j \ge b~. \end{aligned}$$

(3.12)

Here, we have thrown out the continuous variables with non-positive coefficients and we have partitioned the integer variables based on the value of their fractional part $f_j$ being greater than or less than the fractional part $f_0$ of the right-hand side b. Next, rewrite the left hand side of (3.12) as $w + z \ge b$ where w and z are defined as

$$\begin{aligned} w:= & {} \sum _{j:f_j \le f_0} \lceil a_j \rceil x_j + \sum _{j:f_j> f_0} \lfloor a_j \rfloor x_j~, \text{ and } \\ z:= & {} \sum _{j:g_j> 0} g_j y_j + \sum _{j:f_j > f_0} (1 - f_j) x_j~. \end{aligned}$$

Since $w \in \mathbb {Z}$, $z \in \mathbb {R}_+$, and $w + z \ge b$, we can apply inequality (3.9) from Lemma 3.1. This gives $w + \frac{1}{1-f_0} z \ge \lceil b \rceil$ or

$$ \sum _{j:f_j \le f_0} \lceil a_j \rceil x_j + \sum _{j:f_j> f_0} \left( \lfloor a_j \rfloor + \frac{1 - f_j}{1 - f_0} \right) x_j + \frac{1}{1 - f_0}\sum _{j:g_j > 0} g_j y_j \ge \lceil b \rceil ~. $$

Note that $\lfloor a_j \rfloor + \frac{1 - f_j}{1 - f_0} = \lceil a_j \rceil - \frac{f_j-f_0}{1-f_0}$ for all $j \in \{1,\dots ,n\}:f_j > 0$. Note also that for those integer variables j whose fractional part $f_j$ is greater than the fractional part $f_0$ of the right-hand side, the relaxed coefficient $\lceil a_j \rceil - \frac{(f_j-f_0)^+}{1-f_0}$ in the MIR cut (3.11) is stronger than for those j with $f_j \le f_0$.    $\square$