Abstract
A common challenge faced by liner operators in practice is to effectively allocate empty containers now in a way that minimizes the expectation of costs and reduces inefficiencies in the future with uncertainty. To incorporate uncertainties in the operational model, we formulate a two-stage stochastic programming model for the stochastic empty container repositioning (ECR) problem. This paper proposes a separable piecewise linear learning algorithm (SPELL) to approximate the expected cost function. The core of SPELL involves learning steps that provide information for updating the expected cost function adaptively through a sequence of piecewise linear separable approximations. Moreover, SPELL can utilize the network structure of the ECR problem and does not require any information about the distribution of the uncertain parameters. For the two-stage stochastic programs, we prove the convergence of SPELL. Computational results show that SPELL performs well in terms of operating costs. When the scale of the problem is very large and the dimensionality of the problem is increased, SPELL continues to provide consistent performance very efficiently and exhibits excellent convergence performance.
Highlights
Since our empty container repositioning (ECR) problem has a network structure, when we use a separable approximation and the approximation function is piecewise, we can solve the problem as a pure network flow problem, which is a well-known polynomial solvable problem. erefore, the large-scale stochastic ECR problem can be efficiently solved, and we show the efficiency of our approach numerically in this study
In order to incorporate uncertainties in the ECR problem, we build a two-stage stochastic model with uncertain demand, supply, and ship capacity, and this model could be applied to the shipping industry
We propose a separable piecewise linear learning algorithm which does not require any information about the distribution of uncertain parameters, and the learning steps can provide information for updating the expected cost function adaptively by using sample information on the objective function itself
Summary
Our problem focuses on the operational level decisions of the ECR problem. Due to the global trade imbalance, the empty containers in surplus regions like North America have to be repositioned to deficit regions like Asia. If the separable function Q i(x1) is linear or piecewise linear, and the expected recourse function in equation (8) can be replaced with Q (x1) in equation (17), the stochastic ECR problem can be solved as a pure network flow problem, which is a well-known polynomial solvable problem According to their advantage and disadvantage mentioned above, we combine equations (8), (16), and (17) to form an approximation of the expected operational cost function at iteration k as follows: min f x1 cT1 x + Q 0. We sample a realization of random parameters (supply αtp,v(ω), demand βtp,v(ω), and remaining capacity of vessels csr,t(ω), ω ∈ Ω) solve the corresponding deterministic problem, and calculate the stochastic subgradient of the expected cost function, gk. E proposition holds because the approximation functions are not updated when SPELL performs the projection steps. us, if the newly obtained solution xk1+1 produced by Step 6 can pass the locally optimum detector, it can help SPELL to jump out from a local optimum and proceed to the update process
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