A robust ranking of maritime connectivity: revisiting UNCTAD’s liner shipping connectivity index (LSCI)

Abstract

Maritime connectivity is becoming increasingly important due to its inextricable relation to maritime trade, especially in the global economy. This paper presents a robust framework to rank countries on their maritime connectivity. Typically, in most index creation models, the arithmetic mean is used to build a composite index. In this paper, we propose a more reliable ranking approach, by applying the method of stochastic multicriteria acceptability analysis (SMAA). In this, we take into account the variability in the weights assigned to the connectivity factors, thus producing a probabilistic ranking for each country (i.e., the probability to achieve a certain rank). UNCTAD’s ranking, based simply on the arithmetic mean, where all the weights are equal, becomes a special case of our approach. Our results show that China achieves the highest level of connectivity, followed by Singapore.

Appendix: SMAA

Denote the set of alternatives by $A = \left\{ {a_{1} , a_{2} , \ldots ,a_{m} } \right\}$ and the set of criteria by $G = \left\{ {g_{1} , g_{2} , \ldots ,g_{n} } \right\}$. Then, after assigning nonnegative weights $w_{i}$ to each criterion $g_{i}$, with $w_{1} + w_{2} + ... + w_{n} = 1$, the overall evaluation for alternative $a_{k}$ is set as follows:

$$u\left( {a_{k} ,w} \right) = \mathop \sum \limits_{i = 1}^{n} w_{i} g_{i} \left( {a_{k} } \right)$$

There may be imprecisions with respect to the weights assigned to the criteria and the resulting evaluations. To address these issues, SMAA considers two probability distributions $f_{W} \left( w \right)$ and $f_{X} \left( \xi \right)$ on the spaces $W$ and $X,$ respectively. These spaces are defined as follows:

$$W = \left\{ {\left( {w_{1} ,w_{2} ,...,w_{n} } \right) \in {\mathbb{R}}^{n} :w_{i} \ge 0,\forall i = 1,2,...,n;w_{1} + w_{2} + ... + w_{n} = 1} \right\}$$

$X$ is the evaluation space, i.e., the space of the values that a criterion $g_{i} \in G$ can take.

Using SMAA, we first introduce a ranking function relative to alternative $a_{k}$:

$${\text{rank}}\left( {k,\xi ,w} \right) = 1 + \mathop \sum \limits_{h \ne k} \rho \left( {u\left( {\xi_{h} ,w} \right) > u\left( {\xi_{k} ,w} \right)} \right)$$

where $\rho \left( {{\text{false}}} \right) = 0$ and $\rho \left( {{\text{true}}} \right) = 1.$ Then, for every alternative $a_{h} ,$ for every evaluation of the alternatives $\xi \in X$ and for every rank $r = 1,2,...,l,$ the SMAA computes the set of weights of criteria for which alternative $a_{k}$ achieves rank $r$:

$$W_{k}^{r} \left( \xi \right) = \left\{ {w \in W:{\text{rank}}\left( {k,\xi ,w} \right) = r} \right\}$$

The SMAA is used for the computation of the following indices:

Rank acceptability index: This is defined as the measure of the set of weight vectors and evaluations on the underlined criteria for which alternative $a_{k}$ achieves rank $r$:

$$b_{k}^{r} = \int_{\xi \in X} {f_{X} \left( \xi \right)} \int_{{w \in W_{k}^{r} \left( \xi \right)}} {f_{W} \left( w \right)} {\text{d}}w\;{\text{d}}\xi ,$$

where $b_{k}^{r}$ is the probability of $a_{k}$ getting rank r in the preference ranking.

The central weight vector: This is the barycenter of the set of weight vectors for which alternative $a_{k}$ has the best rank and it exhibits the preferences of the average individual weight vector, giving to $a_{k}$ the best rank. Mathematically, it can be represented as follows:

$$w_{k}^{c} = \frac{1}{{b_{k}^{1} }}\int_{\xi \in X} {f_{X} \left( \xi \right)} \int_{{w \in W^{1} \left( \xi \right)}} {f_{W} \left( w \right)w} \,{\text{d}}w\;{\text{d}}\xi .$$

The confidence factor: It represents the frequency with which an alternative is the most preferred outcome using its central weight vector. It is formulated as follows:

$$p_{k}^{c} = \int_{{\xi \in X:u\left( {\xi_{h} ,w_{k}^{c} } \right) \le u\left( {\xi_{k} ,w_{k}^{c} } \right);\forall h = 1,2,...,l}} {f_{X} \left( \xi \right)} \;{\text{d}}\xi$$

Pairwise winning index: Another important index in SMAA is the pairwise winning index, defined as the frequency that alternative $a_{h}$ is preferred over alternative $a_{k}$ in the space of possible weight vectors and possible evaluations on a single criterion:

$$p_{hk} = \int_{w \in W} {f_{W} \left( w \right)} \int_{{\xi \in X:u\left( {\xi_{k} ,w} \right) \le u\left( {\xi_{h} ,w} \right)}} {f_{X} \left( \xi \right)} {\text{d}}\xi \;{\text{d}}w$$

Next, we recall the multidimensional generalization of the Gini index, defined using the rank acceptability index $b_{k}^{r}$. For this purpose, we first consider the upward cumulative rank acceptability index of position $l,$ where $l = 1, 2, \ldots , n - 1,$ for every $r = 1, 2, \ldots , n - 1$, as the probability that alternative $a_{k }$ has rank $l$ or better, that is:

$$b_{ \ge l}^{k} = \sum\nolimits_{s = 1}^{l} {b_{s}^{k} }$$

We can compute the Gini index of the upward cumulative rank acceptability index of position $l,$ as follows:

$$G^{ \ge l} = \frac{{\sum\nolimits_{h = 1}^{n} {\sum\nolimits_{k = 1}^{n} {|b_{ \ge l}^{h} - b_{ \ge l}^{k} |} } }}{{2n\sum\nolimits_{h = 1}^{n} {b_{ \ge l}^{h} } }}$$

which, in view of the fact that $\sum\nolimits_{h = 1}^{n} {b_{r}^{h} = 1}$ for $r = 1, 2, \ldots , n$, and, in turn, $\sum\nolimits_{h = 1}^{n} {b_{ \ge l}^{h} } = l,$ gives:

$$G^{ \ge l} = \frac{{\sum\nolimits_{h = 1}^{n} {\sum\nolimits_{k = 1}^{n} {|b_{ \ge l}^{h} - b_{ \ge l}^{k} |} } }}{2nl}$$

$G^{ \ge l}$ is the concentration of probability to obtain a rank position $l$ or better among the considered alternatives. Analogously, we can define the index $G^{ \le l}$ that measure the concentration of probability to obtain a rank position $l$ or worse, where $l = 2, 3, \ldots , n,$ as:

$$G^{ \le l} = \frac{{\sum\nolimits_{h = 1}^{n} {\sum\nolimits_{k = 1}^{n} {|b_{ \le l}^{h} - b_{ \le l}^{k} |} } }}{2n(n - l + 1)},{\text{ and}}$$

$$b_{ \le l}^{k} = \sum\nolimits_{s = l}^{n} {b_{s}^{k} }$$

define the downward cumulative rank acceptability index of position $l$ for alternative $a_{k}$. The above concepts are found in Lahdelma et al. (1998), Lahdelma and Salminen (2001), and Greco et al. (2018).