Integrated proactive and reactive strategies for sustainable berth allocation and quay crane assignment under uncertainty

Abstract

The berth allocation and quay crane assignment problem (BACAP) is a complex port operation planning problem susceptible to uncertainties, such as vessel arrival time fluctuation to its estimated time of arrival and maritime markets. For promoting reliability and sustainability of container terminals, this paper addresses the optimization of BACAP under the uncertain vessels’ arrival times and fluctuation of loading and unloading volumes. We propose a proactive BACAP strategy considering minimum recovery cost under uncertainty using a reactive strategy. A stochastic programming model is formulated to minimize the basic cost in the baseline schedule, and the recovery cost in real uncertain scenarios. A two-stage meta-heuristic framework based on GA is developed for solving this problem. Numerical experiments and scenario analysis are conducted to validate the effectiveness of the proposed model and the proposed solution approaches.

Appendix: The neighborhood generation approach

The neighborhood generation approach put forwarded in this paper is similar to the one proposed by Zhen et al. (2011), however, the two following aspects in this paper are different from the latter approach. Firstly, it cannot change their sequence if the vessels arrived before the frozen period point in scenario ω. Secondly, the cost function for evacuating the value of vessels in baseline schedule S^o under uncertain scenario ω is updated by $c_{i}^{1 + } s_{i}^{{\Delta { + }}} (\omega ){ + }c_{i}^{1 - } s_{i}^{\Delta - } (\omega ){ + }c_{i}^{2 + } e_{i}^{{\Delta { + }}} (\omega ){ + }c_{i}^{2 - } e_{i}^{\Delta - } (\omega ) + c_{i}^{3 + } y_{i}^{\Delta + } (\omega ) + c_{i}^{3 - } y_{i}^{\Delta - } (\omega )$. The detailed process is shown as follow.

• Step 1: Evaluate the adjusting cost of baseline schedule under uncertain scenario ω ($RCost(v) = c_{i}^{1 + } s_{i}^{{\Delta { + }}} (\omega ){ + }c_{i}^{1 - } s_{i}^{\Delta - } (\omega ){ + }c_{i}^{2 + } e_{i}^{{\Delta { + }}} (\omega ){ + }c_{i}^{2 - } e_{i}^{\Delta - } (\omega ) + c_{i}^{3 + } y_{i}^{\Delta + } (\omega ) + c_{i}^{3 - } y_{i}^{\Delta - } (\omega )$), if their estimated arrival time ETA_i is behind the frozen period point λ(ω);

• Step 2: Calculate the gap of recovery cost between consecutive vessels $Gap_{i} = Rcost(V_{i + 1} ) - Rcost(V_{i} )$, where V_i is the ith vessel in baseline schedule S^o;

• Step 3: Select the vessel V_i with maximum recovery cost gap (Gap_i) to exchange the sequence with their consecutive vessel V_i+1, and generate the first neighborhood (vessel berthing sequence);

• Step 4: Successively exchange the sequence of vessel V_i with 2th, 3th, …, Rth recovery cost gaps with their consecutive vessel V_i+1, and generate the new neighborhoods. If the neighborhood generate has already existed, exchange the vessel V_i with (R + 1)th recovery cost gap with their consecutive vessel V_i+1 until obtain the Rth neighborhood.